Physics Book - User contributions [en]

Vectors

2016-04-16T17:00:33Z

Asatapathy3:

Written by Elizabeth Robelo

Improved by Aparajita Satapathy

In physics, a vector is an object with a magnitude and a direction.

==The Main Idea==

A vector is an object with a magnitude and a direction. The magnitude of a vector is a scalar which represents the length of the vector. A vector is represented by an arrow. The orientation of the vector represents its direction. When a vector is drawn, the starting point is the tail and the ending point is called the head of the vector. Refer to the image below for a visual representation:

[[File:Mathinsight.png|300px|thumb|center|Visual Representation]]

We can also add and subtract vectors. To add two vectors you align them head to tail and make sure that the tails of both vectors coincide with each other. The new added vector is the connecting arrow starting from the tail of one to the head of the other vector.

[[File:Addingvectors.jpg|275px|thumb|center|Adding vector A to B]]



To subtract two vectors reverse the direction of the vector you want to subtract and continue to add them like shown before.



[[File:Subtractingvectors.jpg|350px|thumb|center|Subtracting vector B from A]]

A vector can also be multiplied by a scalar. To multiply a vector by a scalar we can strech, compress or reverse the direction of a vector. If the scalar is between 0 and 1, the vector will be compresseed. If the scalar is greater than 1, the vector will get streched. If the scalar is a negative number, then the vector reverses its direction.

[[File:Ibguides.png|400px|thumb|center|Scalar Multiplication]]

===A Mathematical Model===

Vectors are given by x, y, and z coordinates. They are written in the form <x, y, z> or (xi + yj - zk).
Magnitude: <math> |A| = \sqrt{x^2 + y^2 + z^2} </math>

Dot Product: <math>\mathbf{A}\cdot\mathbf{B}=\sum_{i=1}^n A_iB_i=A_1B_1+A_2B_2+\cdots+A_nB_n</math>

Addition of two vectors: <a1, a2, a3> + <b1, b2, b3> = <a1 + b1, a2 + b2, a3 + b3>

Unit Vector: :<math alt= "u-hat equals the vector u divided by its length">\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}</math>

===A Computational Model===

In Physics 1, we have seen vectors in a computational model when we model it in vpython. Here is a sample code that creates a vector from a baseball to a tennis ball. Notice that the tail begins from the baseball and the head points to the tennis ball. [https://trinket.io/embed/glowscript/a8e75ad500 vectors]

==Examples==

===Simple===
Which of the following statements is correct?

[[File:simproblem.jpg|275px|thumb|center]]

1. <math>\overrightarrow{c} = \overrightarrow{a} + \overrightarrow{b}</math>

2. <math>\overrightarrow{a} = \overrightarrow{b} - \overrightarrow{c}</math>

3. <math>\overrightarrow{a} = \overrightarrow{c} + \overrightarrow{b}</math>

4. <math>\overrightarrow{b} = \overrightarrow{c} + \overrightarrow{a}</math>

The answer is option number 2.

===Middling===
What is the magnitude of the vector C = A - B if A = <10, 5, 8> and B = <9, 4, 3>?

First we need to find the vector C:
A - B = <math> <(10-9), (5-4), (8-3)> = <1, 1, 5> = C</math>

<math>\sqrt{(1)^2 + 1^2 + 5^2} = 5.196</math>

===Difficult===
What is the unit vector in the direction of the vector <10, 5, 8>?
First you have to find the magnitude of the vector given:
<math>\sqrt{10^2 + (5)^2 + 8^2} = 13.74</math>

Finally divide the vector by its magnitude to get the unit vector:

<math>\tfrac{<10, 5, 8>}{13.74}</math>

= <.727, .364, .582>
Notice that the magnitude of the unit vector is equal to 1

==Connectedness==
Vectors will be used in many applications in physics. These can be two dimenstional or three dimensional vectors. Vectors are used to represent forces, fields and momentum.

==History==

In 1835, Giusto Bellavitis abstracted the basic idea of a vector while establishing the concept of equipollence. He called any pair of line segments of the same length and orientation equipollent (meaning equal length). He found a relationship and created the first set of vectors.

William Rowan Hamilton devised the name "vector" as part of his system of quaternions consisting of three dimentional vectors.

Several other mathematicians developed similar vector systems to those of Bellavitis and Hamilton in the 19th century. The system used by Herman Grassman is the one that is most similar to the one used today.

== See also ==

===External links===
Here is a link on more mathematical computations on vectors:
[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[http://mathinsight.org/vector_introduction]

[[Category:Which Category did you place this in?]]

Vectors

2016-04-16T16:59:56Z

Asatapathy3:

Written by Elizabeth Robelo

In physics, a vector is an object with a magnitude and a direction.

==The Main Idea==

A vector is an object with a magnitude and a direction. The magnitude of a vector is a scalar which represents the length of the vector. A vector is represented by an arrow. The orientation of the vector represents its direction. When a vector is drawn, the starting point is the tail and the ending point is called the head of the vector. Refer to the image below for a visual representation:

[[File:Mathinsight.png|300px|thumb|center|Visual Representation]]

We can also add and subtract vectors. To add two vectors you align them head to tail and make sure that the tails of both vectors coincide with each other. The new added vector is the connecting arrow starting from the tail of one to the head of the other vector.

[[File:Addingvectors.jpg|275px|thumb|center|Adding vector A to B]]



To subtract two vectors reverse the direction of the vector you want to subtract and continue to add them like shown before.



[[File:Subtractingvectors.jpg|350px|thumb|center|Subtracting vector B from A]]

A vector can also be multiplied by a scalar. To multiply a vector by a scalar we can strech, compress or reverse the direction of a vector. If the scalar is between 0 and 1, the vector will be compresseed. If the scalar is greater than 1, the vector will get streched. If the scalar is a negative number, then the vector reverses its direction.

[[File:Ibguides.png|400px|thumb|center|Scalar Multiplication]]

===A Mathematical Model===

Vectors are given by x, y, and z coordinates. They are written in the form <x, y, z> or (xi + yj - zk).
Magnitude: <math> |A| = \sqrt{x^2 + y^2 + z^2} </math>

Dot Product: <math>\mathbf{A}\cdot\mathbf{B}=\sum_{i=1}^n A_iB_i=A_1B_1+A_2B_2+\cdots+A_nB_n</math>

Addition of two vectors: <a1, a2, a3> + <b1, b2, b3> = <a1 + b1, a2 + b2, a3 + b3>

Unit Vector: :<math alt= "u-hat equals the vector u divided by its length">\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}</math>

===A Computational Model===

In Physics 1, we have seen vectors in a computational model when we model it in vpython. Here is a sample code that creates a vector from a baseball to a tennis ball. Notice that the tail begins from the baseball and the head points to the tennis ball. [https://trinket.io/embed/glowscript/a8e75ad500 vectors]

==Examples==

===Simple===
Which of the following statements is correct?

[[File:simproblem.jpg|275px|thumb|center]]

1. <math>\overrightarrow{c} = \overrightarrow{a} + \overrightarrow{b}</math>

2. <math>\overrightarrow{a} = \overrightarrow{b} - \overrightarrow{c}</math>

3. <math>\overrightarrow{a} = \overrightarrow{c} + \overrightarrow{b}</math>

4. <math>\overrightarrow{b} = \overrightarrow{c} + \overrightarrow{a}</math>

The answer is option number 2.

===Middling===
What is the magnitude of the vector C = A - B if A = <10, 5, 8> and B = <9, 4, 3>?

First we need to find the vector C:
A - B = <math> <(10-9), (5-4), (8-3)> = <1, 1, 5> = C</math>

<math>\sqrt{(1)^2 + 1^2 + 5^2} = 5.196</math>

===Difficult===
What is the unit vector in the direction of the vector <10, 5, 8>?
First you have to find the magnitude of the vector given:
<math>\sqrt{10^2 + (5)^2 + 8^2} = 13.74</math>

Finally divide the vector by its magnitude to get the unit vector:

<math>\tfrac{<10, 5, 8>}{13.74}</math>

= <.727, .364, .582>
Notice that the magnitude of the unit vector is equal to 1

==Connectedness==
Vectors will be used in many applications in physics. These can be two dimenstional or three dimensional vectors. Vectors are used to represent forces, fields and momentum.

==History==

In 1835, Giusto Bellavitis abstracted the basic idea of a vector while establishing the concept of equipollence. He called any pair of line segments of the same length and orientation equipollent (meaning equal length). He found a relationship and created the first set of vectors.

William Rowan Hamilton devised the name "vector" as part of his system of quaternions consisting of three dimentional vectors.

Several other mathematicians developed similar vector systems to those of Bellavitis and Hamilton in the 19th century. The system used by Herman Grassman is the one that is most similar to the one used today.

== See also ==

===External links===
Here is a link on more mathematical computations on vectors:
[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[http://mathinsight.org/vector_introduction]

[[Category:Which Category did you place this in?]]

Vectors

2016-04-16T16:59:33Z

Asatapathy3: /* History */

Written by Elizabeth Robelo

Improved by: Aparajita Satapathy

In physics, a vector is an object with a magnitude and a direction.

==The Main Idea==

A vector is an object with a magnitude and a direction. The magnitude of a vector is a scalar which represents the length of the vector. A vector is represented by an arrow. The orientation of the vector represents its direction. When a vector is drawn, the starting point is the tail and the ending point is called the head of the vector. Refer to the image below for a visual representation:

[[File:Mathinsight.png|300px|thumb|center|Visual Representation]]

We can also add and subtract vectors. To add two vectors you align them head to tail and make sure that the tails of both vectors coincide with each other. The new added vector is the connecting arrow starting from the tail of one to the head of the other vector.

[[File:Addingvectors.jpg|275px|thumb|center|Adding vector A to B]]



To subtract two vectors reverse the direction of the vector you want to subtract and continue to add them like shown before.



[[File:Subtractingvectors.jpg|350px|thumb|center|Subtracting vector B from A]]

A vector can also be multiplied by a scalar. To multiply a vector by a scalar we can strech, compress or reverse the direction of a vector. If the scalar is between 0 and 1, the vector will be compresseed. If the scalar is greater than 1, the vector will get streched. If the scalar is a negative number, then the vector reverses its direction.

[[File:Ibguides.png|400px|thumb|center|Scalar Multiplication]]

===A Mathematical Model===

Vectors are given by x, y, and z coordinates. They are written in the form <x, y, z> or (xi + yj - zk).
Magnitude: <math> |A| = \sqrt{x^2 + y^2 + z^2} </math>

Dot Product: <math>\mathbf{A}\cdot\mathbf{B}=\sum_{i=1}^n A_iB_i=A_1B_1+A_2B_2+\cdots+A_nB_n</math>

Addition of two vectors: <a1, a2, a3> + <b1, b2, b3> = <a1 + b1, a2 + b2, a3 + b3>

Unit Vector: :<math alt= "u-hat equals the vector u divided by its length">\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}</math>

===A Computational Model===

In Physics 1, we have seen vectors in a computational model when we model it in vpython. Here is a sample code that creates a vector from a baseball to a tennis ball. Notice that the tail begins from the baseball and the head points to the tennis ball. [https://trinket.io/embed/glowscript/a8e75ad500 vectors]

==Examples==

===Simple===
Which of the following statements is correct?

[[File:simproblem.jpg|275px|thumb|center]]

1. <math>\overrightarrow{c} = \overrightarrow{a} + \overrightarrow{b}</math>

2. <math>\overrightarrow{a} = \overrightarrow{b} - \overrightarrow{c}</math>

3. <math>\overrightarrow{a} = \overrightarrow{c} + \overrightarrow{b}</math>

4. <math>\overrightarrow{b} = \overrightarrow{c} + \overrightarrow{a}</math>

The answer is option number 2.

===Middling===
What is the magnitude of the vector C = A - B if A = <10, 5, 8> and B = <9, 4, 3>?

First we need to find the vector C:
A - B = <math> <(10-9), (5-4), (8-3)> = <1, 1, 5> = C</math>

<math>\sqrt{(1)^2 + 1^2 + 5^2} = 5.196</math>

===Difficult===
What is the unit vector in the direction of the vector <10, 5, 8>?
First you have to find the magnitude of the vector given:
<math>\sqrt{10^2 + (5)^2 + 8^2} = 13.74</math>

Finally divide the vector by its magnitude to get the unit vector:

<math>\tfrac{<10, 5, 8>}{13.74}</math>

= <.727, .364, .582>
Notice that the magnitude of the unit vector is equal to 1

==Connectedness==
Vectors will be used in many applications in physics. These can be two dimenstional or three dimensional vectors. Vectors are used to represent forces, fields and momentum.

==History==

In 1835, Giusto Bellavitis abstracted the basic idea of a vector while establishing the concept of equipollence. He called any pair of line segments of the same length and orientation equipollent (meaning equal length). He found a relationship and created the first set of vectors.

William Rowan Hamilton devised the name "vector" as part of his system of quaternions consisting of three dimentional vectors.

Several other mathematicians developed similar vector systems to those of Bellavitis and Hamilton in the 19th century. The system used by Herman Grassman is the one that is most similar to the one used today.

== See also ==

===External links===
Here is a link on more mathematical computations on vectors:
[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[http://mathinsight.org/vector_introduction]

[[Category:Which Category did you place this in?]]

Vectors

2016-04-16T16:54:46Z

Asatapathy3: /* References */

Written by Elizabeth Robelo

Improved by: Aparajita Satapathy

In physics, a vector is an object with a magnitude and a direction.

==The Main Idea==

A vector is an object with a magnitude and a direction. The magnitude of a vector is a scalar which represents the length of the vector. A vector is represented by an arrow. The orientation of the vector represents its direction. When a vector is drawn, the starting point is the tail and the ending point is called the head of the vector. Refer to the image below for a visual representation:

[[File:Mathinsight.png|300px|thumb|center|Visual Representation]]

We can also add and subtract vectors. To add two vectors you align them head to tail and make sure that the tails of both vectors coincide with each other. The new added vector is the connecting arrow starting from the tail of one to the head of the other vector.

[[File:Addingvectors.jpg|275px|thumb|center|Adding vector A to B]]



To subtract two vectors reverse the direction of the vector you want to subtract and continue to add them like shown before.



[[File:Subtractingvectors.jpg|350px|thumb|center|Subtracting vector B from A]]

A vector can also be multiplied by a scalar. To multiply a vector by a scalar we can strech, compress or reverse the direction of a vector. If the scalar is between 0 and 1, the vector will be compresseed. If the scalar is greater than 1, the vector will get streched. If the scalar is a negative number, then the vector reverses its direction.

[[File:Ibguides.png|400px|thumb|center|Scalar Multiplication]]

===A Mathematical Model===

Vectors are given by x, y, and z coordinates. They are written in the form <x, y, z> or (xi + yj - zk).
Magnitude: <math> |A| = \sqrt{x^2 + y^2 + z^2} </math>

Dot Product: <math>\mathbf{A}\cdot\mathbf{B}=\sum_{i=1}^n A_iB_i=A_1B_1+A_2B_2+\cdots+A_nB_n</math>

Addition of two vectors: <a1, a2, a3> + <b1, b2, b3> = <a1 + b1, a2 + b2, a3 + b3>

Unit Vector: :<math alt= "u-hat equals the vector u divided by its length">\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}</math>

===A Computational Model===

In Physics 1, we have seen vectors in a computational model when we model it in vpython. Here is a sample code that creates a vector from a baseball to a tennis ball. Notice that the tail begins from the baseball and the head points to the tennis ball. [https://trinket.io/embed/glowscript/a8e75ad500 vectors]

==Examples==

===Simple===
Which of the following statements is correct?

[[File:simproblem.jpg|275px|thumb|center]]

1. <math>\overrightarrow{c} = \overrightarrow{a} + \overrightarrow{b}</math>

2. <math>\overrightarrow{a} = \overrightarrow{b} - \overrightarrow{c}</math>

3. <math>\overrightarrow{a} = \overrightarrow{c} + \overrightarrow{b}</math>

4. <math>\overrightarrow{b} = \overrightarrow{c} + \overrightarrow{a}</math>

The answer is option number 2.

===Middling===
What is the magnitude of the vector C = A - B if A = <10, 5, 8> and B = <9, 4, 3>?

First we need to find the vector C:
A - B = <math> <(10-9), (5-4), (8-3)> = <1, 1, 5> = C</math>

<math>\sqrt{(1)^2 + 1^2 + 5^2} = 5.196</math>

===Difficult===
What is the unit vector in the direction of the vector <10, 5, 8>?
First you have to find the magnitude of the vector given:
<math>\sqrt{10^2 + (5)^2 + 8^2} = 13.74</math>

Finally divide the vector by its magnitude to get the unit vector:

<math>\tfrac{<10, 5, 8>}{13.74}</math>

= <.727, .364, .582>
Notice that the magnitude of the unit vector is equal to 1

==Connectedness==
Vectors will be used in many applications in physics. These can be two dimenstional or three dimensional vectors. Vectors are used to represent forces, fields and momentum.

==History==

Giusto Bellavitis abstracted the basic idea of a vector in 1835 when he established the concept of equipollence. He called any pair of line segments of the same length and orientation equipollent. He found a relationship and created the first set of vectors.

The name vector was given to us by William Rowan Hamilton as part of his system of quaternions. The vectors he used were three dimensional.

Several other mathematicians developed similar vector systems to those of Bellavitis and Hamilton in the 19th century. The system used by Herman Grassman is the one that is most similar to the one used today. He thought of ideas similar to the cross product and vector differentiation.

== See also ==

===External links===
Here is a link on more mathematical computations on vectors:
[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[http://mathinsight.org/vector_introduction]

[[Category:Which Category did you place this in?]]

Vectors

2016-04-16T16:54:14Z

Asatapathy3: /* External links */

Written by Elizabeth Robelo

Improved by: Aparajita Satapathy

In physics, a vector is an object with a magnitude and a direction.

==The Main Idea==

A vector is an object with a magnitude and a direction. The magnitude of a vector is a scalar which represents the length of the vector. A vector is represented by an arrow. The orientation of the vector represents its direction. When a vector is drawn, the starting point is the tail and the ending point is called the head of the vector. Refer to the image below for a visual representation:

[[File:Mathinsight.png|300px|thumb|center|Visual Representation]]

We can also add and subtract vectors. To add two vectors you align them head to tail and make sure that the tails of both vectors coincide with each other. The new added vector is the connecting arrow starting from the tail of one to the head of the other vector.

[[File:Addingvectors.jpg|275px|thumb|center|Adding vector A to B]]



To subtract two vectors reverse the direction of the vector you want to subtract and continue to add them like shown before.



[[File:Subtractingvectors.jpg|350px|thumb|center|Subtracting vector B from A]]

A vector can also be multiplied by a scalar. To multiply a vector by a scalar we can strech, compress or reverse the direction of a vector. If the scalar is between 0 and 1, the vector will be compresseed. If the scalar is greater than 1, the vector will get streched. If the scalar is a negative number, then the vector reverses its direction.

[[File:Ibguides.png|400px|thumb|center|Scalar Multiplication]]

===A Mathematical Model===

Vectors are given by x, y, and z coordinates. They are written in the form <x, y, z> or (xi + yj - zk).
Magnitude: <math> |A| = \sqrt{x^2 + y^2 + z^2} </math>

Dot Product: <math>\mathbf{A}\cdot\mathbf{B}=\sum_{i=1}^n A_iB_i=A_1B_1+A_2B_2+\cdots+A_nB_n</math>

Addition of two vectors: <a1, a2, a3> + <b1, b2, b3> = <a1 + b1, a2 + b2, a3 + b3>

Unit Vector: :<math alt= "u-hat equals the vector u divided by its length">\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}</math>

===A Computational Model===

In Physics 1, we have seen vectors in a computational model when we model it in vpython. Here is a sample code that creates a vector from a baseball to a tennis ball. Notice that the tail begins from the baseball and the head points to the tennis ball. [https://trinket.io/embed/glowscript/a8e75ad500 vectors]

==Examples==

===Simple===
Which of the following statements is correct?

[[File:simproblem.jpg|275px|thumb|center]]

1. <math>\overrightarrow{c} = \overrightarrow{a} + \overrightarrow{b}</math>

2. <math>\overrightarrow{a} = \overrightarrow{b} - \overrightarrow{c}</math>

3. <math>\overrightarrow{a} = \overrightarrow{c} + \overrightarrow{b}</math>

4. <math>\overrightarrow{b} = \overrightarrow{c} + \overrightarrow{a}</math>

The answer is option number 2.

===Middling===
What is the magnitude of the vector C = A - B if A = <10, 5, 8> and B = <9, 4, 3>?

First we need to find the vector C:
A - B = <math> <(10-9), (5-4), (8-3)> = <1, 1, 5> = C</math>

<math>\sqrt{(1)^2 + 1^2 + 5^2} = 5.196</math>

===Difficult===
What is the unit vector in the direction of the vector <10, 5, 8>?
First you have to find the magnitude of the vector given:
<math>\sqrt{10^2 + (5)^2 + 8^2} = 13.74</math>

Finally divide the vector by its magnitude to get the unit vector:

<math>\tfrac{<10, 5, 8>}{13.74}</math>

= <.727, .364, .582>
Notice that the magnitude of the unit vector is equal to 1

==Connectedness==
Vectors will be used in many applications in physics. These can be two dimenstional or three dimensional vectors. Vectors are used to represent forces, fields and momentum.

==History==

Giusto Bellavitis abstracted the basic idea of a vector in 1835 when he established the concept of equipollence. He called any pair of line segments of the same length and orientation equipollent. He found a relationship and created the first set of vectors.

The name vector was given to us by William Rowan Hamilton as part of his system of quaternions. The vectors he used were three dimensional.

Several other mathematicians developed similar vector systems to those of Bellavitis and Hamilton in the 19th century. The system used by Herman Grassman is the one that is most similar to the one used today. He thought of ideas similar to the cross product and vector differentiation.

== See also ==

===External links===
Here is a link on more mathematical computations on vectors:
[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction]

[[Category:Which Category did you place this in?]]

Vectors

2016-04-16T16:53:45Z

Asatapathy3: /*Connectedness */

Written by Elizabeth Robelo

Improved by: Aparajita Satapathy

In physics, a vector is an object with a magnitude and a direction.

==The Main Idea==

A vector is an object with a magnitude and a direction. The magnitude of a vector is a scalar which represents the length of the vector. A vector is represented by an arrow. The orientation of the vector represents its direction. When a vector is drawn, the starting point is the tail and the ending point is called the head of the vector. Refer to the image below for a visual representation:

[[File:Mathinsight.png|300px|thumb|center|Visual Representation]]

We can also add and subtract vectors. To add two vectors you align them head to tail and make sure that the tails of both vectors coincide with each other. The new added vector is the connecting arrow starting from the tail of one to the head of the other vector.

[[File:Addingvectors.jpg|275px|thumb|center|Adding vector A to B]]



To subtract two vectors reverse the direction of the vector you want to subtract and continue to add them like shown before.



[[File:Subtractingvectors.jpg|350px|thumb|center|Subtracting vector B from A]]

A vector can also be multiplied by a scalar. To multiply a vector by a scalar we can strech, compress or reverse the direction of a vector. If the scalar is between 0 and 1, the vector will be compresseed. If the scalar is greater than 1, the vector will get streched. If the scalar is a negative number, then the vector reverses its direction.

[[File:Ibguides.png|400px|thumb|center|Scalar Multiplication]]

===A Mathematical Model===

Vectors are given by x, y, and z coordinates. They are written in the form <x, y, z> or (xi + yj - zk).
Magnitude: <math> |A| = \sqrt{x^2 + y^2 + z^2} </math>

Dot Product: <math>\mathbf{A}\cdot\mathbf{B}=\sum_{i=1}^n A_iB_i=A_1B_1+A_2B_2+\cdots+A_nB_n</math>

Addition of two vectors: <a1, a2, a3> + <b1, b2, b3> = <a1 + b1, a2 + b2, a3 + b3>

Unit Vector: :<math alt= "u-hat equals the vector u divided by its length">\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}</math>

===A Computational Model===

In Physics 1, we have seen vectors in a computational model when we model it in vpython. Here is a sample code that creates a vector from a baseball to a tennis ball. Notice that the tail begins from the baseball and the head points to the tennis ball. [https://trinket.io/embed/glowscript/a8e75ad500 vectors]

==Examples==

===Simple===
Which of the following statements is correct?

[[File:simproblem.jpg|275px|thumb|center]]

1. <math>\overrightarrow{c} = \overrightarrow{a} + \overrightarrow{b}</math>

2. <math>\overrightarrow{a} = \overrightarrow{b} - \overrightarrow{c}</math>

3. <math>\overrightarrow{a} = \overrightarrow{c} + \overrightarrow{b}</math>

4. <math>\overrightarrow{b} = \overrightarrow{c} + \overrightarrow{a}</math>

The answer is option number 2.

===Middling===
What is the magnitude of the vector C = A - B if A = <10, 5, 8> and B = <9, 4, 3>?

First we need to find the vector C:
A - B = <math> <(10-9), (5-4), (8-3)> = <1, 1, 5> = C</math>

<math>\sqrt{(1)^2 + 1^2 + 5^2} = 5.196</math>

===Difficult===
What is the unit vector in the direction of the vector <10, 5, 8>?
First you have to find the magnitude of the vector given:
<math>\sqrt{10^2 + (5)^2 + 8^2} = 13.74</math>

Finally divide the vector by its magnitude to get the unit vector:

<math>\tfrac{<10, 5, 8>}{13.74}</math>

= <.727, .364, .582>
Notice that the magnitude of the unit vector is equal to 1

==Connectedness==
Vectors will be used in many applications in physics. These can be two dimenstional or three dimensional vectors. Vectors are used to represent forces, fields and momentum.

==History==

Giusto Bellavitis abstracted the basic idea of a vector in 1835 when he established the concept of equipollence. He called any pair of line segments of the same length and orientation equipollent. He found a relationship and created the first set of vectors.

The name vector was given to us by William Rowan Hamilton as part of his system of quaternions. The vectors he used were three dimensional.

Several other mathematicians developed similar vector systems to those of Bellavitis and Hamilton in the 19th century. The system used by Herman Grassman is the one that is most similar to the one used today. He thought of ideas similar to the cross product and vector differentiation.

== See also ==

===External links===
[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction]

[[Category:Which Category did you place this in?]]

Vectors

2016-04-16T16:50:19Z

Asatapathy3: /* Examples */

Written by Elizabeth Robelo

Improved by: Aparajita Satapathy

In physics, a vector is an object with a magnitude and a direction.

==The Main Idea==

A vector is an object with a magnitude and a direction. The magnitude of a vector is a scalar which represents the length of the vector. A vector is represented by an arrow. The orientation of the vector represents its direction. When a vector is drawn, the starting point is the tail and the ending point is called the head of the vector. Refer to the image below for a visual representation:

[[File:Mathinsight.png|300px|thumb|center|Visual Representation]]

We can also add and subtract vectors. To add two vectors you align them head to tail and make sure that the tails of both vectors coincide with each other. The new added vector is the connecting arrow starting from the tail of one to the head of the other vector.

[[File:Addingvectors.jpg|275px|thumb|center|Adding vector A to B]]



To subtract two vectors reverse the direction of the vector you want to subtract and continue to add them like shown before.



[[File:Subtractingvectors.jpg|350px|thumb|center|Subtracting vector B from A]]

A vector can also be multiplied by a scalar. To multiply a vector by a scalar we can strech, compress or reverse the direction of a vector. If the scalar is between 0 and 1, the vector will be compresseed. If the scalar is greater than 1, the vector will get streched. If the scalar is a negative number, then the vector reverses its direction.

[[File:Ibguides.png|400px|thumb|center|Scalar Multiplication]]

===A Mathematical Model===

Vectors are given by x, y, and z coordinates. They are written in the form <x, y, z> or (xi + yj - zk).
Magnitude: <math> |A| = \sqrt{x^2 + y^2 + z^2} </math>

Dot Product: <math>\mathbf{A}\cdot\mathbf{B}=\sum_{i=1}^n A_iB_i=A_1B_1+A_2B_2+\cdots+A_nB_n</math>

Addition of two vectors: <a1, a2, a3> + <b1, b2, b3> = <a1 + b1, a2 + b2, a3 + b3>

Unit Vector: :<math alt= "u-hat equals the vector u divided by its length">\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}</math>

===A Computational Model===

In Physics 1, we have seen vectors in a computational model when we model it in vpython. Here is a sample code that creates a vector from a baseball to a tennis ball. Notice that the tail begins from the baseball and the head points to the tennis ball. [https://trinket.io/embed/glowscript/a8e75ad500 vectors]

==Examples==

===Simple===
Which of the following statements is correct?

[[File:simproblem.jpg|275px|thumb|center]]

1. <math>\overrightarrow{c} = \overrightarrow{a} + \overrightarrow{b}</math>

2. <math>\overrightarrow{a} = \overrightarrow{b} - \overrightarrow{c}</math>

3. <math>\overrightarrow{a} = \overrightarrow{c} + \overrightarrow{b}</math>

4. <math>\overrightarrow{b} = \overrightarrow{c} + \overrightarrow{a}</math>

The answer is option number 2.

===Middling===
What is the magnitude of the vector C = A - B if A = <10, 5, 8> and B = <9, 4, 3>?

First we need to find the vector C:
A - B = <math> <(10-9), (5-4), (8-3)> = <1, 1, 5> = C</math>

<math>\sqrt{(1)^2 + 1^2 + 5^2} = 5.196</math>

===Difficult===
What is the unit vector in the direction of the vector <10, 5, 8>?
First you have to find the magnitude of the vector given:
<math>\sqrt{10^2 + (5)^2 + 8^2} = 13.74</math>

Finally divide the vector by its magnitude to get the unit vector:

<math>\tfrac{<10, 5, 8>}{13.74}</math>

= <.727, .364, .582>
Notice that the magnitude of the unit vector is equal to 1

==Applications==
You will use vectors for everything in physics ranging from velocity to gravitational field.

==History==

Giusto Bellavitis abstracted the basic idea of a vector in 1835 when he established the concept of equipollence. He called any pair of line segments of the same length and orientation equipollent. He found a relationship and created the first set of vectors.

The name vector was given to us by William Rowan Hamilton as part of his system of quaternions. The vectors he used were three dimensional.

Several other mathematicians developed similar vector systems to those of Bellavitis and Hamilton in the 19th century. The system used by Herman Grassman is the one that is most similar to the one used today. He thought of ideas similar to the cross product and vector differentiation.

== See also ==

===External links===
[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction]

[[Category:Which Category did you place this in?]]

Vectors

2016-04-16T16:39:29Z

Asatapathy3: /* A Computational Model */

Written by Elizabeth Robelo

Improved by: Aparajita Satapathy

In physics, a vector is an object with a magnitude and a direction.

==The Main Idea==

A vector is an object with a magnitude and a direction. The magnitude of a vector is a scalar which represents the length of the vector. A vector is represented by an arrow. The orientation of the vector represents its direction. When a vector is drawn, the starting point is the tail and the ending point is called the head of the vector. Refer to the image below for a visual representation:

[[File:Mathinsight.png|300px|thumb|center|Visual Representation]]

We can also add and subtract vectors. To add two vectors you align them head to tail and make sure that the tails of both vectors coincide with each other. The new added vector is the connecting arrow starting from the tail of one to the head of the other vector.

[[File:Addingvectors.jpg|275px|thumb|center|Adding vector A to B]]



To subtract two vectors reverse the direction of the vector you want to subtract and continue to add them like shown before.



[[File:Subtractingvectors.jpg|350px|thumb|center|Subtracting vector B from A]]

A vector can also be multiplied by a scalar. To multiply a vector by a scalar we can strech, compress or reverse the direction of a vector. If the scalar is between 0 and 1, the vector will be compresseed. If the scalar is greater than 1, the vector will get streched. If the scalar is a negative number, then the vector reverses its direction.

[[File:Ibguides.png|400px|thumb|center|Scalar Multiplication]]

===A Mathematical Model===

Vectors are given by x, y, and z coordinates. They are written in the form <x, y, z> or (xi + yj - zk).
Magnitude: <math> |A| = \sqrt{x^2 + y^2 + z^2} </math>

Dot Product: <math>\mathbf{A}\cdot\mathbf{B}=\sum_{i=1}^n A_iB_i=A_1B_1+A_2B_2+\cdots+A_nB_n</math>

Addition of two vectors: <a1, a2, a3> + <b1, b2, b3> = <a1 + b1, a2 + b2, a3 + b3>

Unit Vector: :<math alt= "u-hat equals the vector u divided by its length">\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}</math>

===A Computational Model===

In Physics 1, we have seen vectors in a computational model when we model it in vpython. Here is a sample code that creates a vector from a baseball to a tennis ball. Notice that the tail begins from the baseball and the head points to the tennis ball. [https://trinket.io/embed/glowscript/a8e75ad500 vectors]

==Examples==

===Simple===
Which of the following statements is correct?

[[File:simproblem.jpg|275px|thumb|center]]

1. <math>\overrightarrow{c} = \overrightarrow{b} + \overrightarrow{a}</math>

2. <math>\overrightarrow{a} = \overrightarrow{b} - \overrightarrow{c}</math>

3. <math>\overrightarrow{a} = \overrightarrow{b} + \overrightarrow{c}</math>

4. <math>\overrightarrow{b} = \overrightarrow{a} + \overrightarrow{c}</math>

The answer is 2.

===Middling===
What is the magnitude of the vector C = A - B if A = <6, 21, 17> and B = <12, 7, 15>?

A - B = <math> <6-12, 21-7, 17-15> = <-6, 14, 2> = C</math>

<math>\sqrt{(-6)^2 + 14^2 + 2^2} = 15.36</math>

===Difficult===
What is the unit vector in the direction of the vector <12, -15, 9>?
First you have to find the magnitude of the vector given:
<math>\sqrt{12^2 + (-15)^2 + 9^2} = 21.21</math>

Finally divide the vector by its magnitude to get the unit vector:

<math>\tfrac{<12,-15,9>}{180}</math>

= <.565, -.707, .424>

==Applications==
You will use vectors for everything in physics ranging from velocity to gravitational field.

==History==

Giusto Bellavitis abstracted the basic idea of a vector in 1835 when he established the concept of equipollence. He called any pair of line segments of the same length and orientation equipollent. He found a relationship and created the first set of vectors.

The name vector was given to us by William Rowan Hamilton as part of his system of quaternions. The vectors he used were three dimensional.

Several other mathematicians developed similar vector systems to those of Bellavitis and Hamilton in the 19th century. The system used by Herman Grassman is the one that is most similar to the one used today. He thought of ideas similar to the cross product and vector differentiation.

== See also ==

===External links===
[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction]

[[Category:Which Category did you place this in?]]

Vectors

2016-04-16T16:36:23Z

Asatapathy3: /* A Mathematical Model */

Written by Elizabeth Robelo

Improved by: Aparajita Satapathy

In physics, a vector is an object with a magnitude and a direction.

==The Main Idea==

A vector is an object with a magnitude and a direction. The magnitude of a vector is a scalar which represents the length of the vector. A vector is represented by an arrow. The orientation of the vector represents its direction. When a vector is drawn, the starting point is the tail and the ending point is called the head of the vector. Refer to the image below for a visual representation:

[[File:Mathinsight.png|300px|thumb|center|Visual Representation]]

We can also add and subtract vectors. To add two vectors you align them head to tail and make sure that the tails of both vectors coincide with each other. The new added vector is the connecting arrow starting from the tail of one to the head of the other vector.

[[File:Addingvectors.jpg|275px|thumb|center|Adding vector A to B]]



To subtract two vectors reverse the direction of the vector you want to subtract and continue to add them like shown before.



[[File:Subtractingvectors.jpg|350px|thumb|center|Subtracting vector B from A]]

A vector can also be multiplied by a scalar. To multiply a vector by a scalar we can strech, compress or reverse the direction of a vector. If the scalar is between 0 and 1, the vector will be compresseed. If the scalar is greater than 1, the vector will get streched. If the scalar is a negative number, then the vector reverses its direction.

[[File:Ibguides.png|400px|thumb|center|Scalar Multiplication]]

===A Mathematical Model===

Vectors are given by x, y, and z coordinates. They are written in the form <x, y, z> or (xi + yj - zk).
Magnitude: <math> |A| = \sqrt{x^2 + y^2 + z^2} </math>

Dot Product: <math>\mathbf{A}\cdot\mathbf{B}=\sum_{i=1}^n A_iB_i=A_1B_1+A_2B_2+\cdots+A_nB_n</math>

Addition of two vectors: <a1, a2, a3> + <b1, b2, b3> = <a1 + b1, a2 + b2, a3 + b3>

Unit Vector: :<math alt= "u-hat equals the vector u divided by its length">\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}</math>

===A Computational Model===

Here you have vpython code creating a vector from one object to another. [https://trinket.io/embed/glowscript/a8e75ad500 vectors]

==Examples==

===Simple===
Which of the following statements is correct?

[[File:simproblem.jpg|275px|thumb|center]]

1. <math>\overrightarrow{c} = \overrightarrow{b} + \overrightarrow{a}</math>

2. <math>\overrightarrow{a} = \overrightarrow{b} - \overrightarrow{c}</math>

3. <math>\overrightarrow{a} = \overrightarrow{b} + \overrightarrow{c}</math>

4. <math>\overrightarrow{b} = \overrightarrow{a} + \overrightarrow{c}</math>

The answer is 2.

===Middling===
What is the magnitude of the vector C = A - B if A = <6, 21, 17> and B = <12, 7, 15>?

A - B = <math> <6-12, 21-7, 17-15> = <-6, 14, 2> = C</math>

<math>\sqrt{(-6)^2 + 14^2 + 2^2} = 15.36</math>

===Difficult===
What is the unit vector in the direction of the vector <12, -15, 9>?
First you have to find the magnitude of the vector given:
<math>\sqrt{12^2 + (-15)^2 + 9^2} = 21.21</math>

Finally divide the vector by its magnitude to get the unit vector:

<math>\tfrac{<12,-15,9>}{180}</math>

= <.565, -.707, .424>

==Applications==
You will use vectors for everything in physics ranging from velocity to gravitational field.

==History==

Giusto Bellavitis abstracted the basic idea of a vector in 1835 when he established the concept of equipollence. He called any pair of line segments of the same length and orientation equipollent. He found a relationship and created the first set of vectors.

The name vector was given to us by William Rowan Hamilton as part of his system of quaternions. The vectors he used were three dimensional.

Several other mathematicians developed similar vector systems to those of Bellavitis and Hamilton in the 19th century. The system used by Herman Grassman is the one that is most similar to the one used today. He thought of ideas similar to the cross product and vector differentiation.

== See also ==

===External links===
[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction]

[[Category:Which Category did you place this in?]]

Vectors

2016-04-16T16:35:39Z

Asatapathy3: /* A Mathematical Model */

Written by Elizabeth Robelo

Improved by: Aparajita Satapathy

In physics, a vector is an object with a magnitude and a direction.

==The Main Idea==

A vector is an object with a magnitude and a direction. The magnitude of a vector is a scalar which represents the length of the vector. A vector is represented by an arrow. The orientation of the vector represents its direction. When a vector is drawn, the starting point is the tail and the ending point is called the head of the vector. Refer to the image below for a visual representation:

[[File:Mathinsight.png|300px|thumb|center|Visual Representation]]

We can also add and subtract vectors. To add two vectors you align them head to tail and make sure that the tails of both vectors coincide with each other. The new added vector is the connecting arrow starting from the tail of one to the head of the other vector.

[[File:Addingvectors.jpg|275px|thumb|center|Adding vector A to B]]



To subtract two vectors reverse the direction of the vector you want to subtract and continue to add them like shown before.



[[File:Subtractingvectors.jpg|350px|thumb|center|Subtracting vector B from A]]

A vector can also be multiplied by a scalar. To multiply a vector by a scalar we can strech, compress or reverse the direction of a vector. If the scalar is between 0 and 1, the vector will be compresseed. If the scalar is greater than 1, the vector will get streched. If the scalar is a negative number, then the vector reverses its direction.

[[File:Ibguides.png|400px|thumb|center|Scalar Multiplication]]

===A Mathematical Model===

Vectors are given by x, y, and z coordinates. They are written in the form <x, y, z> or xi + yj - zk.
Magnitude: <math> |A| = \sqrt{x^2 + y^2 + z^2} </math>

Dot Product: <math>\mathbf{A}\cdot\mathbf{B}=\sum_{i=1}^n A_iB_i=A_1B_1+A_2B_2+\cdots+A_nB_n</math>
:<math>
<x1, x2, x3> \cdot <y1, y2, y3> &= (x1)(y1) + (x2)(y2) + (x3)(y3) \\
\end{align}
</math>

Addition of two vectors: <a1, a2, a3> + <b1, b2, b3> = <a1 + b1, a2 + b2, a3 + b3>

Unit Vector: :<math alt= "u-hat equals the vector u divided by its length">\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}</math>

===A Computational Model===

Here you have vpython code creating a vector from one object to another. [https://trinket.io/embed/glowscript/a8e75ad500 vectors]

==Examples==

===Simple===
Which of the following statements is correct?

[[File:simproblem.jpg|275px|thumb|center]]

1. <math>\overrightarrow{c} = \overrightarrow{b} + \overrightarrow{a}</math>

2. <math>\overrightarrow{a} = \overrightarrow{b} - \overrightarrow{c}</math>

3. <math>\overrightarrow{a} = \overrightarrow{b} + \overrightarrow{c}</math>

4. <math>\overrightarrow{b} = \overrightarrow{a} + \overrightarrow{c}</math>

The answer is 2.

===Middling===
What is the magnitude of the vector C = A - B if A = <6, 21, 17> and B = <12, 7, 15>?

A - B = <math> <6-12, 21-7, 17-15> = <-6, 14, 2> = C</math>

<math>\sqrt{(-6)^2 + 14^2 + 2^2} = 15.36</math>

===Difficult===
What is the unit vector in the direction of the vector <12, -15, 9>?
First you have to find the magnitude of the vector given:
<math>\sqrt{12^2 + (-15)^2 + 9^2} = 21.21</math>

Finally divide the vector by its magnitude to get the unit vector:

<math>\tfrac{<12,-15,9>}{180}</math>

= <.565, -.707, .424>

==Applications==
You will use vectors for everything in physics ranging from velocity to gravitational field.

==History==

Giusto Bellavitis abstracted the basic idea of a vector in 1835 when he established the concept of equipollence. He called any pair of line segments of the same length and orientation equipollent. He found a relationship and created the first set of vectors.

The name vector was given to us by William Rowan Hamilton as part of his system of quaternions. The vectors he used were three dimensional.

Several other mathematicians developed similar vector systems to those of Bellavitis and Hamilton in the 19th century. The system used by Herman Grassman is the one that is most similar to the one used today. He thought of ideas similar to the cross product and vector differentiation.

== See also ==

===External links===
[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction]

[[Category:Which Category did you place this in?]]

Vectors

2016-04-16T16:34:53Z

Asatapathy3: /* A Mathematical Model */

Written by Elizabeth Robelo

Improved by: Aparajita Satapathy

In physics, a vector is an object with a magnitude and a direction.

==The Main Idea==

A vector is an object with a magnitude and a direction. The magnitude of a vector is a scalar which represents the length of the vector. A vector is represented by an arrow. The orientation of the vector represents its direction. When a vector is drawn, the starting point is the tail and the ending point is called the head of the vector. Refer to the image below for a visual representation:

[[File:Mathinsight.png|300px|thumb|center|Visual Representation]]

We can also add and subtract vectors. To add two vectors you align them head to tail and make sure that the tails of both vectors coincide with each other. The new added vector is the connecting arrow starting from the tail of one to the head of the other vector.

[[File:Addingvectors.jpg|275px|thumb|center|Adding vector A to B]]



To subtract two vectors reverse the direction of the vector you want to subtract and continue to add them like shown before.



[[File:Subtractingvectors.jpg|350px|thumb|center|Subtracting vector B from A]]

A vector can also be multiplied by a scalar. To multiply a vector by a scalar we can strech, compress or reverse the direction of a vector. If the scalar is between 0 and 1, the vector will be compresseed. If the scalar is greater than 1, the vector will get streched. If the scalar is a negative number, then the vector reverses its direction.

[[File:Ibguides.png|400px|thumb|center|Scalar Multiplication]]

===A Mathematical Model===

Vectors are given by x, y, and z coordinates. They are written in the form <x, y, z> or xi + yj - zk.
Magnitude: <math> |A| = \sqrt{x^2 + y^2 + z^2} </math>

Dot Product: <math>\mathbf{A}\cdot\mathbf{B}=\sum_{i=1}^n A_iB_i=A_1B_1+A_2B_2+\cdots+A_nB_n</math>
:<math>
\begin{align}
\ <x1, x2, x3> \cdot <y1, y2, y3> &= (x1)(y1) + (x2)(y2) + (x3)(y3) \\
\end{align}
</math>

Addition of two vectors: <a1, a2, a3> + <b1, b2, b3> = <a1 + b1, a2 + b2, a3 + b3>

Unit Vector: :<math alt= "u-hat equals the vector u divided by its length">\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}</math>

===A Computational Model===

Here you have vpython code creating a vector from one object to another. [https://trinket.io/embed/glowscript/a8e75ad500 vectors]

==Examples==

===Simple===
Which of the following statements is correct?

[[File:simproblem.jpg|275px|thumb|center]]

1. <math>\overrightarrow{c} = \overrightarrow{b} + \overrightarrow{a}</math>

2. <math>\overrightarrow{a} = \overrightarrow{b} - \overrightarrow{c}</math>

3. <math>\overrightarrow{a} = \overrightarrow{b} + \overrightarrow{c}</math>

4. <math>\overrightarrow{b} = \overrightarrow{a} + \overrightarrow{c}</math>

The answer is 2.

===Middling===
What is the magnitude of the vector C = A - B if A = <6, 21, 17> and B = <12, 7, 15>?

A - B = <math> <6-12, 21-7, 17-15> = <-6, 14, 2> = C</math>

<math>\sqrt{(-6)^2 + 14^2 + 2^2} = 15.36</math>

===Difficult===
What is the unit vector in the direction of the vector <12, -15, 9>?
First you have to find the magnitude of the vector given:
<math>\sqrt{12^2 + (-15)^2 + 9^2} = 21.21</math>

Finally divide the vector by its magnitude to get the unit vector:

<math>\tfrac{<12,-15,9>}{180}</math>

= <.565, -.707, .424>

==Applications==
You will use vectors for everything in physics ranging from velocity to gravitational field.

==History==

Giusto Bellavitis abstracted the basic idea of a vector in 1835 when he established the concept of equipollence. He called any pair of line segments of the same length and orientation equipollent. He found a relationship and created the first set of vectors.

The name vector was given to us by William Rowan Hamilton as part of his system of quaternions. The vectors he used were three dimensional.

Several other mathematicians developed similar vector systems to those of Bellavitis and Hamilton in the 19th century. The system used by Herman Grassman is the one that is most similar to the one used today. He thought of ideas similar to the cross product and vector differentiation.

== See also ==

===External links===
[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction]

[[Category:Which Category did you place this in?]]

Vectors

2016-04-16T16:29:06Z

Asatapathy3:

Written by Elizabeth Robelo

Improved by: Aparajita Satapathy

In physics, a vector is an object with a magnitude and a direction.

==The Main Idea==

A vector is an object with a magnitude and a direction. The magnitude of a vector is a scalar which represents the length of the vector. A vector is represented by an arrow. The orientation of the vector represents its direction. When a vector is drawn, the starting point is the tail and the ending point is called the head of the vector. Refer to the image below for a visual representation:

[[File:Mathinsight.png|300px|thumb|center|Visual Representation]]

We can also add and subtract vectors. To add two vectors you align them head to tail and make sure that the tails of both vectors coincide with each other. The new added vector is the connecting arrow starting from the tail of one to the head of the other vector.

[[File:Addingvectors.jpg|275px|thumb|center|Adding vector A to B]]



To subtract two vectors reverse the direction of the vector you want to subtract and continue to add them like shown before.



[[File:Subtractingvectors.jpg|350px|thumb|center|Subtracting vector B from A]]

A vector can also be multiplied by a scalar. To multiply a vector by a scalar we can strech, compress or reverse the direction of a vector. If the scalar is between 0 and 1, the vector will be compresseed. If the scalar is greater than 1, the vector will get streched. If the scalar is a negative number, then the vector reverses its direction.

[[File:Ibguides.png|400px|thumb|center|Scalar Multiplication]]

===A Mathematical Model===

Vectors are given by x, y, and z coordinates. They are written in the form <x, y, z> or xi + yj - zk.
Magnitude: <math> |A| = \sqrt{x^2 + y^2 + z^2} </math>

Dot Product: <math>\mathbf{A}\cdot\mathbf{B}=\sum_{i=1}^n A_iB_i=A_1B_1+A_2B_2+\cdots+A_nB_n</math>

Addition of two vectors: <a1, a2, a3> + <b1, b2, b3> = <a1 + b1, a2 + b2, a3 + b3>

Unit Vector: :<math alt= "u-hat equals the vector u divided by its length">\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}</math>

===A Computational Model===

Here you have vpython code creating a vector from one object to another. [https://trinket.io/embed/glowscript/a8e75ad500 vectors]

==Examples==

===Simple===
Which of the following statements is correct?

[[File:simproblem.jpg|275px|thumb|center]]

1. <math>\overrightarrow{c} = \overrightarrow{b} + \overrightarrow{a}</math>

2. <math>\overrightarrow{a} = \overrightarrow{b} - \overrightarrow{c}</math>

3. <math>\overrightarrow{a} = \overrightarrow{b} + \overrightarrow{c}</math>

4. <math>\overrightarrow{b} = \overrightarrow{a} + \overrightarrow{c}</math>

The answer is 2.

===Middling===
What is the magnitude of the vector C = A - B if A = <6, 21, 17> and B = <12, 7, 15>?

A - B = <math> <6-12, 21-7, 17-15> = <-6, 14, 2> = C</math>

<math>\sqrt{(-6)^2 + 14^2 + 2^2} = 15.36</math>

===Difficult===
What is the unit vector in the direction of the vector <12, -15, 9>?
First you have to find the magnitude of the vector given:
<math>\sqrt{12^2 + (-15)^2 + 9^2} = 21.21</math>

Finally divide the vector by its magnitude to get the unit vector:

<math>\tfrac{<12,-15,9>}{180}</math>

= <.565, -.707, .424>

==Applications==
You will use vectors for everything in physics ranging from velocity to gravitational field.

==History==

Giusto Bellavitis abstracted the basic idea of a vector in 1835 when he established the concept of equipollence. He called any pair of line segments of the same length and orientation equipollent. He found a relationship and created the first set of vectors.

The name vector was given to us by William Rowan Hamilton as part of his system of quaternions. The vectors he used were three dimensional.

Several other mathematicians developed similar vector systems to those of Bellavitis and Hamilton in the 19th century. The system used by Herman Grassman is the one that is most similar to the one used today. He thought of ideas similar to the cross product and vector differentiation.

== See also ==

===External links===
[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction]

[[Category:Which Category did you place this in?]]

Vectors

2016-04-16T16:27:11Z

Asatapathy3: /* Connectedness */

Written by Elizabeth Robelo

Improved by: Aparajita Satapathy

In physics, a vector is an object with a magnitude and a direction.

==The Main Idea==

A vector is an object with a magnitude and a direction. The magnitude of a vector is a scalar which represents the length of the vector. A vector is represented by an arrow. The orientation of the vector represents its direction. When a vector is drawn, the starting point is the tail and the ending point is called the head of the vector. Refer to the image below for a visual representation:

[[File:Mathinsight.png|300px|thumb|center|Visual Representation]]

We can also add and subtract vectors. To add two vectors you align them head to tail and make sure that the tails of both vectors coincide with each other. The new added vector is the connecting arrow starting from the tail of one to the head of the other vector.

[[File:Addingvectors.jpg|275px|thumb|center|Adding vector A to B]]



To subtract two vectors reverse the direction of the vector you want to subtract and continue to add them like shown before.



[[File:Subtractingvectors.jpg|350px|thumb|center|Subtracting vector B from A]]

A vector can also be multiplied by a scalar. To multiply a vector by a scalar we can strech, compress or reverse the direction of a vector. If the scalar is between 0 and 1, the vector will be compresseed. If the scalar is greater than 1, the vector will get streched. If the scalar is a negative number, then the vector reverses its direction.

[[File:Ibguides.png|400px|thumb|center|Scalar Multiplication]]

A unit vector is a vector that points in the same direction as the original vector with magnitude 1. We usually designate the unit vector with a "hat" <math alt= r-hat>{\hat{\imath}}</math>. The unit vector is often called the normal vector.

===A Mathematical Model===

Vectors are given by x, y, and z coordinates. They are written in the form <x, y, z> or xi + yj - zk.
Magnitude: <math> |A| = \sqrt{x^2 + y^2 + z^2} </math>

Addition of two vectors: <a1, a2, a3> + <b1, b2, b3> = <a1 + b1, a2 + b2, a3 + b3>

Unit Vector: :<math alt= "u-hat equals the vector u divided by its length">\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}</math>

===A Computational Model===

Here you have vpython code creating a vector from one object to another. [https://trinket.io/embed/glowscript/a8e75ad500 vectors]

==Examples==

===Simple===
Which of the following statements is correct?

[[File:simproblem.jpg|275px|thumb|center]]

1. <math>\overrightarrow{c} = \overrightarrow{b} + \overrightarrow{a}</math>

2. <math>\overrightarrow{a} = \overrightarrow{b} - \overrightarrow{c}</math>

3. <math>\overrightarrow{a} = \overrightarrow{b} + \overrightarrow{c}</math>

4. <math>\overrightarrow{b} = \overrightarrow{a} + \overrightarrow{c}</math>

The answer is 2.

===Middling===
What is the magnitude of the vector C = A - B if A = <6, 21, 17> and B = <12, 7, 15>?

A - B = <math> <6-12, 21-7, 17-15> = <-6, 14, 2> = C</math>

<math>\sqrt{(-6)^2 + 14^2 + 2^2} = 15.36</math>

===Difficult===
What is the unit vector in the direction of the vector <12, -15, 9>?
First you have to find the magnitude of the vector given:
<math>\sqrt{12^2 + (-15)^2 + 9^2} = 21.21</math>

Finally divide the vector by its magnitude to get the unit vector:

<math>\tfrac{<12,-15,9>}{180}</math>

= <.565, -.707, .424>

==Connectedness==
You will use vectors for everything in physics ranging from velocity to gravitational field.

==History==

Giusto Bellavitis abstracted the basic idea of a vector in 1835 when he established the concept of equipollence. He called any pair of line segments of the same length and orientation equipollent. He found a relationship and created the first set of vectors.

The name vector was given to us by William Rowan Hamilton as part of his system of quaternions. The vectors he used were three dimensional.

Several other mathematicians developed similar vector systems to those of Bellavitis and Hamilton in the 19th century. The system used by Herman Grassman is the one that is most similar to the one used today. He thought of ideas similar to the cross product and vector differentiation.

== See also ==

===External links===
[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction]

[[Category:Which Category did you place this in?]]

File:Ibguides.png

2016-04-16T16:22:55Z

Asatapathy3:

Vectors

2016-04-16T16:22:40Z

Asatapathy3:

Written by Elizabeth Robelo

Improved by: Aparajita Satapathy

In physics, a vector is an object with a magnitude and a direction.

==The Main Idea==

A vector is an object with a magnitude and a direction. The magnitude of a vector is a scalar which represents the length of the vector. A vector is represented by an arrow. The orientation of the vector represents its direction. When a vector is drawn, the starting point is the tail and the ending point is called the head of the vector. Refer to the image below for a visual representation:

[[File:Mathinsight.png|300px|thumb|center|Visual Representation]]

We can also add and subtract vectors. To add two vectors you align them head to tail and make sure that the tails of both vectors coincide with each other. The new added vector is the connecting arrow starting from the tail of one to the head of the other vector.

[[File:Addingvectors.jpg|275px|thumb|center|Adding vector A to B]]



To subtract two vectors reverse the direction of the vector you want to subtract and continue to add them like shown before.



[[File:Subtractingvectors.jpg|350px|thumb|center|Subtracting vector B from A]]

A vector can also be multiplied by a scalar. To multiply a vector by a scalar we can strech, compress or reverse the direction of a vector. If the scalar is between 0 and 1, the vector will be compresseed. If the scalar is greater than 1, the vector will get streched. If the scalar is a negative number, then the vector reverses its direction.

[[File:Ibguides.png|400px|thumb|center|Scalar Multiplication]]

A unit vector is a vector that points in the same direction as the original vector with magnitude 1. We usually designate the unit vector with a "hat" <math alt= r-hat>{\hat{\imath}}</math>. The unit vector is often called the normal vector.

===A Mathematical Model===

Vectors are given by x, y, and z coordinates. They are written in the form <x, y, z> or xi + yj - zk.
Magnitude: <math> |A| = \sqrt{x^2 + y^2 + z^2} </math>

Addition of two vectors: <a1, a2, a3> + <b1, b2, b3> = <a1 + b1, a2 + b2, a3 + b3>

Unit Vector: :<math alt= "u-hat equals the vector u divided by its length">\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}</math>

===A Computational Model===

Here you have vpython code creating a vector from one object to another. [https://trinket.io/embed/glowscript/a8e75ad500 vectors]

==Examples==

===Simple===
Which of the following statements is correct?

[[File:simproblem.jpg|275px|thumb|center]]

1. <math>\overrightarrow{c} = \overrightarrow{b} + \overrightarrow{a}</math>

2. <math>\overrightarrow{a} = \overrightarrow{b} - \overrightarrow{c}</math>

3. <math>\overrightarrow{a} = \overrightarrow{b} + \overrightarrow{c}</math>

4. <math>\overrightarrow{b} = \overrightarrow{a} + \overrightarrow{c}</math>

The answer is 2.

===Middling===
What is the magnitude of the vector C = A - B if A = <6, 21, 17> and B = <12, 7, 15>?

A - B = <math> <6-12, 21-7, 17-15> = <-6, 14, 2> = C</math>

<math>\sqrt{(-6)^2 + 14^2 + 2^2} = 15.36</math>

===Difficult===
What is the unit vector in the direction of the vector <12, -15, 9>?
First you have to find the magnitude of the vector given:
<math>\sqrt{12^2 + (-15)^2 + 9^2} = 21.21</math>

Finally divide the vector by its magnitude to get the unit vector:

<math>\tfrac{<12,-15,9>}{180}</math>

= <.565, -.707, .424>

==Applications==
You will use vectors for everything in physics ranging from velocity to gravitational field.

==History==

Giusto Bellavitis abstracted the basic idea of a vector in 1835 when he established the concept of equipollence. He called any pair of line segments of the same length and orientation equipollent. He found a relationship and created the first set of vectors.

The name vector was given to us by William Rowan Hamilton as part of his system of quaternions. The vectors he used were three dimensional.

Several other mathematicians developed similar vector systems to those of Bellavitis and Hamilton in the 19th century. The system used by Herman Grassman is the one that is most similar to the one used today. He thought of ideas similar to the cross product and vector differentiation.

== See also ==

===External links===
[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction]

[[Category:Which Category did you place this in?]]

Vectors

2016-04-16T16:21:26Z

Asatapathy3: /* The Main Idea */

Written by Elizabeth Robelo

Improved by: Aparajita Satapathy

In physics, a vector is an object with a magnitude and a direction.

==The Main Idea==

A vector is an object with a magnitude and a direction. The magnitude of a vector is a scalar which represents the length of the vector. A vector is represented by an arrow. The orientation of the vector represents its direction. When a vector is drawn, the starting point is the tail and the ending point is called the head of the vector. Refer to the image below for a visual representation:

[[File:Mathinsight.png|300px|thumb|center|Visual Representation]]

We can also add and subtract vectors. To add two vectors you align them head to tail and make sure that the tails of both vectors coincide with each other. The new added vector is the connecting arrow starting from the tail of one to the head of the other vector.

[[File:Addingvectors.jpg|275px|thumb|center|Adding vector A to B]]



To subtract two vectors reverse the direction of the vector you want to subtract and continue to add them like shown before.



[[File:Subtractingvectors.jpg|350px|thumb|center|Subtracting vector B from A]]

A vector can also be multiplied by a scalar. To multiply a vector by a scalar we can strech, compress or reverse the direction of a vector. If the scalar is between 0 and 1, the vector will be compresseed. If the scalar is greater than 1, the vector will get streched. If the scalar is a negative number, then the vector reverses its direction.

[[File:Ibguides.com|400px|thumb|center|Scalar Multiplication]]

A unit vector is a vector that points in the same direction as the original vector with magnitude 1. We usually designate the unit vector with a "hat" <math alt= r-hat>{\hat{\imath}}</math>. The unit vector is often called the normal vector.

===A Mathematical Model===

Vectors are given by x, y, and z coordinates. They are written in the form <x, y, z> or xi + yj - zk.
Magnitude: <math> |A| = \sqrt{x^2 + y^2 + z^2} </math>

Addition of two vectors: <a1, a2, a3> + <b1, b2, b3> = <a1 + b1, a2 + b2, a3 + b3>

Unit Vector: :<math alt= "u-hat equals the vector u divided by its length">\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}</math>

===A Computational Model===

Here you have vpython code creating a vector from one object to another. [https://trinket.io/embed/glowscript/a8e75ad500 vectors]

==Examples==

===Simple===
Which of the following statements is correct?

[[File:simproblem.jpg|275px|thumb|center]]

1. <math>\overrightarrow{c} = \overrightarrow{b} + \overrightarrow{a}</math>

2. <math>\overrightarrow{a} = \overrightarrow{b} - \overrightarrow{c}</math>

3. <math>\overrightarrow{a} = \overrightarrow{b} + \overrightarrow{c}</math>

4. <math>\overrightarrow{b} = \overrightarrow{a} + \overrightarrow{c}</math>

The answer is 2.

===Middling===
What is the magnitude of the vector C = A - B if A = <6, 21, 17> and B = <12, 7, 15>?

A - B = <math> <6-12, 21-7, 17-15> = <-6, 14, 2> = C</math>

<math>\sqrt{(-6)^2 + 14^2 + 2^2} = 15.36</math>

===Difficult===
What is the unit vector in the direction of the vector <12, -15, 9>?
First you have to find the magnitude of the vector given:
<math>\sqrt{12^2 + (-15)^2 + 9^2} = 21.21</math>

Finally divide the vector by its magnitude to get the unit vector:

<math>\tfrac{<12,-15,9>}{180}</math>

= <.565, -.707, .424>

==Applications==
You will use vectors for everything in physics ranging from velocity to gravitational field.

==History==

Giusto Bellavitis abstracted the basic idea of a vector in 1835 when he established the concept of equipollence. He called any pair of line segments of the same length and orientation equipollent. He found a relationship and created the first set of vectors.

The name vector was given to us by William Rowan Hamilton as part of his system of quaternions. The vectors he used were three dimensional.

Several other mathematicians developed similar vector systems to those of Bellavitis and Hamilton in the 19th century. The system used by Herman Grassman is the one that is most similar to the one used today. He thought of ideas similar to the cross product and vector differentiation.

== See also ==

===External links===
[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction]

[[Category:Which Category did you place this in?]]

File:Mathinsight.png

2016-04-16T16:08:11Z

Asatapathy3:

Vectors

2016-04-16T16:07:13Z

Asatapathy3:

Written by Elizabeth Robelo

Improved by: Aparajita Satapathy

In physics, a vector is an object with a magnitude and a direction.

==The Main Idea==

A vector is an object with a magnitude and a direction. The magnitude of a vector is a scalar which represents the length of the vector. A vector is represented by an arrow. The orientation of the vector represents its direction. When a vector is drawn, the starting point is the tail and the ending point is called the head of the vector. Refer to the image below for a visual representation:

[[File:Mathinsight.png|300px|thumb|center|Visual Representation]]

Vectors can be added and subtracted to each other. To add two vectors you put them head to tail. The connecting arrow starting from the tail of one to the head of the other is the new vector.

[[File:Addingvectors.jpg|275px|thumb|center|Adding vector A to B]]



To subtract two vectors reverse the direction of the one you want to subtract and continue to add them like shown before.



[[File:Subtractingvectors.jpg|350px|thumb|center|Subtracting vector B from A]]

A unit vector is a vector that points in the same direction as the original vector with magnitude 1. We usually designate the unit vector with a "hat" <math alt= r-hat>{\hat{\imath}}</math>. The unit vector is often called the normal vector.

===A Mathematical Model===

Vectors are given by x, y, and z coordinates. They are written in the form <x, y, z> or xi + yj - zk.
Magnitude: <math> |A| = \sqrt{x^2 + y^2 + z^2} </math>

Addition of two vectors: <a1, a2, a3> + <b1, b2, b3> = <a1 + b1, a2 + b2, a3 + b3>

Unit Vector: :<math alt= "u-hat equals the vector u divided by its length">\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}</math>

===A Computational Model===

Here you have vpython code creating a vector from one object to another. [https://trinket.io/embed/glowscript/a8e75ad500 vectors]

==Examples==

===Simple===
Which of the following statements is correct?

[[File:simproblem.jpg|275px|thumb|center]]

1. <math>\overrightarrow{c} = \overrightarrow{b} + \overrightarrow{a}</math>

2. <math>\overrightarrow{a} = \overrightarrow{b} - \overrightarrow{c}</math>

3. <math>\overrightarrow{a} = \overrightarrow{b} + \overrightarrow{c}</math>

4. <math>\overrightarrow{b} = \overrightarrow{a} + \overrightarrow{c}</math>

The answer is 2.

===Middling===
What is the magnitude of the vector C = A - B if A = <6, 21, 17> and B = <12, 7, 15>?

A - B = <math> <6-12, 21-7, 17-15> = <-6, 14, 2> = C</math>

<math>\sqrt{(-6)^2 + 14^2 + 2^2} = 15.36</math>

===Difficult===
What is the unit vector in the direction of the vector <12, -15, 9>?
First you have to find the magnitude of the vector given:
<math>\sqrt{12^2 + (-15)^2 + 9^2} = 21.21</math>

Finally divide the vector by its magnitude to get the unit vector:

<math>\tfrac{<12,-15,9>}{180}</math>

= <.565, -.707, .424>

==Applications==
You will use vectors for everything in physics ranging from velocity to gravitational field.

==History==

Giusto Bellavitis abstracted the basic idea of a vector in 1835 when he established the concept of equipollence. He called any pair of line segments of the same length and orientation equipollent. He found a relationship and created the first set of vectors.

The name vector was given to us by William Rowan Hamilton as part of his system of quaternions. The vectors he used were three dimensional.

Several other mathematicians developed similar vector systems to those of Bellavitis and Hamilton in the 19th century. The system used by Herman Grassman is the one that is most similar to the one used today. He thought of ideas similar to the cross product and vector differentiation.

== See also ==

===External links===
[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction]

[[Category:Which Category did you place this in?]]

Vectors

2016-04-16T16:05:22Z

Asatapathy3: /* The Main Idea */

Written by Elizabeth Robelo

Improved by: Aparajita Satapathy

In physics, a vector is an object with a magnitude and a direction.

==The Main Idea==

A vector is an object with a magnitude and a direction. The magnitude of a vector is a scalar which represents the length of the vector. A vector is represented by an arrow. The orientation of the vector represents its direction. When a vector is drawn, the starting point is the tail and the ending point is called the head of the vector. Refer to the image below for a visual representation:

[[File:Mathinsight.org|300px|thumb|center|Visual Representation]]

Vectors can be added and subtracted to each other. To add two vectors you put them head to tail. The connecting arrow starting from the tail of one to the head of the other is the new vector.

[[File:Addingvectors.jpg|275px|thumb|center|Adding vector A to B]]



To subtract two vectors reverse the direction of the one you want to subtract and continue to add them like shown before.



[[File:Subtractingvectors.jpg|350px|thumb|center|Subtracting vector B from A]]

A unit vector is a vector that points in the same direction as the original vector with magnitude 1. We usually designate the unit vector with a "hat" <math alt= r-hat>{\hat{\imath}}</math>. The unit vector is often called the normal vector.

===A Mathematical Model===

Vectors are given by x, y, and z coordinates. They are written in the form <x, y, z> or xi + yj - zk.
Magnitude: <math> |A| = \sqrt{x^2 + y^2 + z^2} </math>

Addition of two vectors: <a1, a2, a3> + <b1, b2, b3> = <a1 + b1, a2 + b2, a3 + b3>

Unit Vector: :<math alt= "u-hat equals the vector u divided by its length">\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}</math>

===A Computational Model===

Here you have vpython code creating a vector from one object to another. [https://trinket.io/embed/glowscript/a8e75ad500 vectors]

==Examples==

===Simple===
Which of the following statements is correct?

[[File:simproblem.jpg|275px|thumb|center]]

1. <math>\overrightarrow{c} = \overrightarrow{b} + \overrightarrow{a}</math>

2. <math>\overrightarrow{a} = \overrightarrow{b} - \overrightarrow{c}</math>

3. <math>\overrightarrow{a} = \overrightarrow{b} + \overrightarrow{c}</math>

4. <math>\overrightarrow{b} = \overrightarrow{a} + \overrightarrow{c}</math>

The answer is 2.

===Middling===
What is the magnitude of the vector C = A - B if A = <6, 21, 17> and B = <12, 7, 15>?

A - B = <math> <6-12, 21-7, 17-15> = <-6, 14, 2> = C</math>

<math>\sqrt{(-6)^2 + 14^2 + 2^2} = 15.36</math>

===Difficult===
What is the unit vector in the direction of the vector <12, -15, 9>?
First you have to find the magnitude of the vector given:
<math>\sqrt{12^2 + (-15)^2 + 9^2} = 21.21</math>

Finally divide the vector by its magnitude to get the unit vector:

<math>\tfrac{<12,-15,9>}{180}</math>

= <.565, -.707, .424>

==Applications==
You will use vectors for everything in physics ranging from velocity to gravitational field.

==History==

Giusto Bellavitis abstracted the basic idea of a vector in 1835 when he established the concept of equipollence. He called any pair of line segments of the same length and orientation equipollent. He found a relationship and created the first set of vectors.

The name vector was given to us by William Rowan Hamilton as part of his system of quaternions. The vectors he used were three dimensional.

Several other mathematicians developed similar vector systems to those of Bellavitis and Hamilton in the 19th century. The system used by Herman Grassman is the one that is most similar to the one used today. He thought of ideas similar to the cross product and vector differentiation.

== See also ==

===External links===
[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction]

[[Category:Which Category did you place this in?]]

Vectors

2016-04-16T15:59:31Z

Asatapathy3: /* The Main Idea */

Written by Elizabeth Robelo

Improved by: Aparajita Satapathy

In physics, a vector is an object with a magnitude and a direction.

==The Main Idea==

A vector is an object with a magnitude and a direction. The magnitude of a vector is a scalar which represents the length of the vector. A vector is represented by an arrow. The orientation of the vector represents its direction. When a vector is drawn, the starting point is the tail and the ending point is called the head of the vector. Refer to the image below for a visual representation:

Vectors can be added and subtracted to each other. To add two vectors you put them head to tail. The connecting arrow starting from the tail of one to the head of the other is the new vector.

[[File:Addingvectors.jpg|275px|thumb|center|Adding vector A to B]]



To subtract two vectors reverse the direction of the one you want to subtract and continue to add them like shown before.



[[File:Subtractingvectors.jpg|350px|thumb|center|Subtracting vector B from A]]

A unit vector is a vector that points in the same direction as the original vector with magnitude 1. We usually designate the unit vector with a "hat" <math alt= r-hat>{\hat{\imath}}</math>. The unit vector is often called the normal vector.

===A Mathematical Model===

Vectors are given by x, y, and z coordinates. They are written in the form <x, y, z> or xi + yj - zk.
Magnitude: <math> |A| = \sqrt{x^2 + y^2 + z^2} </math>

Addition of two vectors: <a1, a2, a3> + <b1, b2, b3> = <a1 + b1, a2 + b2, a3 + b3>

Unit Vector: :<math alt= "u-hat equals the vector u divided by its length">\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}</math>

===A Computational Model===

Here you have vpython code creating a vector from one object to another. [https://trinket.io/embed/glowscript/a8e75ad500 vectors]

==Examples==

===Simple===
Which of the following statements is correct?

[[File:simproblem.jpg|275px|thumb|center]]

1. <math>\overrightarrow{c} = \overrightarrow{b} + \overrightarrow{a}</math>

2. <math>\overrightarrow{a} = \overrightarrow{b} - \overrightarrow{c}</math>

3. <math>\overrightarrow{a} = \overrightarrow{b} + \overrightarrow{c}</math>

4. <math>\overrightarrow{b} = \overrightarrow{a} + \overrightarrow{c}</math>

The answer is 2.

===Middling===
What is the magnitude of the vector C = A - B if A = <6, 21, 17> and B = <12, 7, 15>?

A - B = <math> <6-12, 21-7, 17-15> = <-6, 14, 2> = C</math>

<math>\sqrt{(-6)^2 + 14^2 + 2^2} = 15.36</math>

===Difficult===
What is the unit vector in the direction of the vector <12, -15, 9>?
First you have to find the magnitude of the vector given:
<math>\sqrt{12^2 + (-15)^2 + 9^2} = 21.21</math>

Finally divide the vector by its magnitude to get the unit vector:

<math>\tfrac{<12,-15,9>}{180}</math>

= <.565, -.707, .424>

==Applications==
You will use vectors for everything in physics ranging from velocity to gravitational field.

==History==

Giusto Bellavitis abstracted the basic idea of a vector in 1835 when he established the concept of equipollence. He called any pair of line segments of the same length and orientation equipollent. He found a relationship and created the first set of vectors.

The name vector was given to us by William Rowan Hamilton as part of his system of quaternions. The vectors he used were three dimensional.

Several other mathematicians developed similar vector systems to those of Bellavitis and Hamilton in the 19th century. The system used by Herman Grassman is the one that is most similar to the one used today. He thought of ideas similar to the cross product and vector differentiation.

== See also ==

===External links===
[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction]

[[Category:Which Category did you place this in?]]

Vectors

2016-04-16T15:58:00Z

Asatapathy3:

Written by Elizabeth Robelo

Improved by: Aparajita Satapathy

In physics, a vector is an object with a magnitude and a direction.

==The Main Idea==

A vector is an object with a magnitude and a direction. The magnitude of a vector is a scalar which represents the length of the vector. A vector is represented by an arrow. The orientation of the vector represents its direction. When a vector is drawn, the starting point is the tail and the ending point is called the head of the vector. Refer to the image below for a visual representation:

[[File:mathinsight.org|media|image|image|vector.png|Visual Representation of a Vector]]

Vectors can be added and subtracted to each other. To add two vectors you put them head to tail. The connecting arrow starting from the tail of one to the head of the other is the new vector.

[[File:Addingvectors.jpg|275px|thumb|center|Adding vector A to B]]



To subtract two vectors reverse the direction of the one you want to subtract and continue to add them like shown before.



[[File:Subtractingvectors.jpg|350px|thumb|center|Subtracting vector B from A]]

A unit vector is a vector that points in the same direction as the original vector with magnitude 1. We usually designate the unit vector with a "hat" <math alt= r-hat>{\hat{\imath}}</math>. The unit vector is often called the normal vector.

===A Mathematical Model===

Vectors are given by x, y, and z coordinates. They are written in the form <x, y, z> or xi + yj - zk.
Magnitude: <math> |A| = \sqrt{x^2 + y^2 + z^2} </math>

Addition of two vectors: <a1, a2, a3> + <b1, b2, b3> = <a1 + b1, a2 + b2, a3 + b3>

Unit Vector: :<math alt= "u-hat equals the vector u divided by its length">\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}</math>

===A Computational Model===

Here you have vpython code creating a vector from one object to another. [https://trinket.io/embed/glowscript/a8e75ad500 vectors]

==Examples==

===Simple===
Which of the following statements is correct?

[[File:simproblem.jpg|275px|thumb|center]]

1. <math>\overrightarrow{c} = \overrightarrow{b} + \overrightarrow{a}</math>

2. <math>\overrightarrow{a} = \overrightarrow{b} - \overrightarrow{c}</math>

3. <math>\overrightarrow{a} = \overrightarrow{b} + \overrightarrow{c}</math>

4. <math>\overrightarrow{b} = \overrightarrow{a} + \overrightarrow{c}</math>

The answer is 2.

===Middling===
What is the magnitude of the vector C = A - B if A = <6, 21, 17> and B = <12, 7, 15>?

A - B = <math> <6-12, 21-7, 17-15> = <-6, 14, 2> = C</math>

<math>\sqrt{(-6)^2 + 14^2 + 2^2} = 15.36</math>

===Difficult===
What is the unit vector in the direction of the vector <12, -15, 9>?
First you have to find the magnitude of the vector given:
<math>\sqrt{12^2 + (-15)^2 + 9^2} = 21.21</math>

Finally divide the vector by its magnitude to get the unit vector:

<math>\tfrac{<12,-15,9>}{180}</math>

= <.565, -.707, .424>

==Applications==
You will use vectors for everything in physics ranging from velocity to gravitational field.

==History==

Giusto Bellavitis abstracted the basic idea of a vector in 1835 when he established the concept of equipollence. He called any pair of line segments of the same length and orientation equipollent. He found a relationship and created the first set of vectors.

The name vector was given to us by William Rowan Hamilton as part of his system of quaternions. The vectors he used were three dimensional.

Several other mathematicians developed similar vector systems to those of Bellavitis and Hamilton in the 19th century. The system used by Herman Grassman is the one that is most similar to the one used today. He thought of ideas similar to the cross product and vector differentiation.

== See also ==

===External links===
[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction]

[[Category:Which Category did you place this in?]]

File:Mathinsight.org-media-image-image-vector.png

2016-04-16T15:55:44Z

Asatapathy3:

Vectors

2016-04-16T15:54:24Z

Asatapathy3:

'

Written by Elizabeth Robelo
Improved by: Aparajita Satapathy

In physics, a vector is an object with a magnitude and a direction.

==The Main Idea==

A vector is an object with a magnitude and a direction. The magnitude of a vector is a scalar which represents the length of the vector. A vector is represented by an arrow. The orientation of the vector represents its direction. When a vector is drawn, the starting point is the tail and the ending point is called the head of the vector. Refer to the image below for a visual representation:

[[File:mathinsight.org|media|image|image|vector.png|Visual Representation of a Vector]]

Vectors can be added and subtracted to each other. To add two vectors you put them head to tail. The connecting arrow starting from the tail of one to the head of the other is the new vector.

[[File:Addingvectors.jpg|275px|thumb|center|Adding vector A to B]]



To subtract two vectors reverse the direction of the one you want to subtract and continue to add them like shown before.



[[File:Subtractingvectors.jpg|350px|thumb|center|Subtracting vector B from A]]

A unit vector is a vector that points in the same direction as the original vector with magnitude 1. We usually designate the unit vector with a "hat" <math alt= r-hat>{\hat{\imath}}</math>. The unit vector is often called the normal vector.

===A Mathematical Model===

Vectors are given by x, y, and z coordinates. They are written in the form <x, y, z> or xi + yj - zk.
Magnitude: <math> |A| = \sqrt{x^2 + y^2 + z^2} </math>

Addition of two vectors: <a1, a2, a3> + <b1, b2, b3> = <a1 + b1, a2 + b2, a3 + b3>

Unit Vector: :<math alt= "u-hat equals the vector u divided by its length">\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}</math>

===A Computational Model===

Here you have vpython code creating a vector from one object to another. [https://trinket.io/embed/glowscript/a8e75ad500 vectors]

==Examples==

===Simple===
Which of the following statements is correct?

[[File:simproblem.jpg|275px|thumb|center]]

1. <math>\overrightarrow{c} = \overrightarrow{b} + \overrightarrow{a}</math>

2. <math>\overrightarrow{a} = \overrightarrow{b} - \overrightarrow{c}</math>

3. <math>\overrightarrow{a} = \overrightarrow{b} + \overrightarrow{c}</math>

4. <math>\overrightarrow{b} = \overrightarrow{a} + \overrightarrow{c}</math>

The answer is 2.

===Middling===
What is the magnitude of the vector C = A - B if A = <6, 21, 17> and B = <12, 7, 15>?

A - B = <math> <6-12, 21-7, 17-15> = <-6, 14, 2> = C</math>

<math>\sqrt{(-6)^2 + 14^2 + 2^2} = 15.36</math>

===Difficult===
What is the unit vector in the direction of the vector <12, -15, 9>?
First you have to find the magnitude of the vector given:
<math>\sqrt{12^2 + (-15)^2 + 9^2} = 21.21</math>

Finally divide the vector by its magnitude to get the unit vector:

<math>\tfrac{<12,-15,9>}{180}</math>

= <.565, -.707, .424>

==Applications==
You will use vectors for everything in physics ranging from velocity to gravitational field.

==History==

Giusto Bellavitis abstracted the basic idea of a vector in 1835 when he established the concept of equipollence. He called any pair of line segments of the same length and orientation equipollent. He found a relationship and created the first set of vectors.

The name vector was given to us by William Rowan Hamilton as part of his system of quaternions. The vectors he used were three dimensional.

Several other mathematicians developed similar vector systems to those of Bellavitis and Hamilton in the 19th century. The system used by Herman Grassman is the one that is most similar to the one used today. He thought of ideas similar to the cross product and vector differentiation.

== See also ==

===External links===
[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction]

[[Category:Which Category did you place this in?]]

Vectors

2016-04-16T15:51:02Z

Asatapathy3:

'

Written by Elizabeth Robelo
Improved by: Aparajita Satapathy

In physics, a vector is an object with a magnitude and a direction.

==The Main Idea==

A vector is an object with a magnitude and a direction. The magnitude of a vector is a scalar which represents the length of the vector. A vector is represented by an arrow. The orientation of the vector represents its direction. When a vector is drawn, the starting point is the tail and the ending point is called the head of the vector. Refer to the image below for a visual representation:

[[File:phatcode.net|res|214|images|vector_i.png|Visual Representation of a Vector]]

Vectors can be added and subtracted to each other. To add two vectors you put them head to tail. The connecting arrow starting from the tail of one to the head of the other is the new vector.

[[File:Addingvectors.jpg|275px|thumb|center|Adding vector A to B]]



To subtract two vectors reverse the direction of the one you want to subtract and continue to add them like shown before.



[[File:Subtractingvectors.jpg|350px|thumb|center|Subtracting vector B from A]]

A unit vector is a vector that points in the same direction as the original vector with magnitude 1. We usually designate the unit vector with a "hat" <math alt= r-hat>{\hat{\imath}}</math>. The unit vector is often called the normal vector.

===A Mathematical Model===

Vectors are given by x, y, and z coordinates. They are written in the form <x, y, z> or xi + yj - zk.
Magnitude: <math> |A| = \sqrt{x^2 + y^2 + z^2} </math>

Addition of two vectors: <a1, a2, a3> + <b1, b2, b3> = <a1 + b1, a2 + b2, a3 + b3>

Unit Vector: :<math alt= "u-hat equals the vector u divided by its length">\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}</math>

===A Computational Model===

Here you have vpython code creating a vector from one object to another. [https://trinket.io/embed/glowscript/a8e75ad500 vectors]

==Examples==

===Simple===
Which of the following statements is correct?

[[File:simproblem.jpg|275px|thumb|center]]

1. <math>\overrightarrow{c} = \overrightarrow{b} + \overrightarrow{a}</math>

2. <math>\overrightarrow{a} = \overrightarrow{b} - \overrightarrow{c}</math>

3. <math>\overrightarrow{a} = \overrightarrow{b} + \overrightarrow{c}</math>

4. <math>\overrightarrow{b} = \overrightarrow{a} + \overrightarrow{c}</math>

The answer is 2.

===Middling===
What is the magnitude of the vector C = A - B if A = <6, 21, 17> and B = <12, 7, 15>?

A - B = <math> <6-12, 21-7, 17-15> = <-6, 14, 2> = C</math>

<math>\sqrt{(-6)^2 + 14^2 + 2^2} = 15.36</math>

===Difficult===
What is the unit vector in the direction of the vector <12, -15, 9>?
First you have to find the magnitude of the vector given:
<math>\sqrt{12^2 + (-15)^2 + 9^2} = 21.21</math>

Finally divide the vector by its magnitude to get the unit vector:

<math>\tfrac{<12,-15,9>}{180}</math>

= <.565, -.707, .424>

==Applications==
You will use vectors for everything in physics ranging from velocity to gravitational field.

==History==

Giusto Bellavitis abstracted the basic idea of a vector in 1835 when he established the concept of equipollence. He called any pair of line segments of the same length and orientation equipollent. He found a relationship and created the first set of vectors.

The name vector was given to us by William Rowan Hamilton as part of his system of quaternions. The vectors he used were three dimensional.

Several other mathematicians developed similar vector systems to those of Bellavitis and Hamilton in the 19th century. The system used by Herman Grassman is the one that is most similar to the one used today. He thought of ideas similar to the cross product and vector differentiation.

== See also ==

===External links===
[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction]

[[Category:Which Category did you place this in?]]

File:Phatcode.net-res-214-images-vector i.png

2016-04-16T15:49:35Z

Asatapathy3:

Vectors

2016-04-16T15:47:41Z

Asatapathy3:

'

Written by Elizabeth Robelo
Improved by: Aparajita Satapathy

In physics, a vector is an object with a magnitude and a direction.

==The Main Idea==

A vector is an object with a magnitude and a direction. The magnitude of a vector is a scalar which represents the length of the vector. A vector is represented by an arrow. The orientation of the vector represents its direction. When a vector is drawn, the starting point is the tail and the ending point is called the head of the vector. Refer to the image below for a visual representation:

[[File:phatcode.net|res|214|images|vector_i.png]]

Vectors can be added and subtracted to each other. To add two vectors you put them head to tail. The connecting arrow starting from the tail of one to the head of the other is the new vector.

[[File:Addingvectors.jpg|275px|thumb|center|Adding vector A to B]]



To subtract two vectors reverse the direction of the one you want to subtract and continue to add them like shown before.



[[File:Subtractingvectors.jpg|350px|thumb|center|Subtracting vector B from A]]

A unit vector is a vector that points in the same direction as the original vector with magnitude 1. We usually designate the unit vector with a "hat" <math alt= r-hat>{\hat{\imath}}</math>. The unit vector is often called the normal vector.

===A Mathematical Model===

Vectors are given by x, y, and z coordinates. They are written in the form <x, y, z> or xi + yj - zk.
Magnitude: <math> |A| = \sqrt{x^2 + y^2 + z^2} </math>

Addition of two vectors: <a1, a2, a3> + <b1, b2, b3> = <a1 + b1, a2 + b2, a3 + b3>

Unit Vector: :<math alt= "u-hat equals the vector u divided by its length">\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}</math>

===A Computational Model===

Here you have vpython code creating a vector from one object to another. [https://trinket.io/embed/glowscript/a8e75ad500 vectors]

==Examples==

===Simple===
Which of the following statements is correct?

[[File:simproblem.jpg|275px|thumb|center]]

1. <math>\overrightarrow{c} = \overrightarrow{b} + \overrightarrow{a}</math>

2. <math>\overrightarrow{a} = \overrightarrow{b} - \overrightarrow{c}</math>

3. <math>\overrightarrow{a} = \overrightarrow{b} + \overrightarrow{c}</math>

4. <math>\overrightarrow{b} = \overrightarrow{a} + \overrightarrow{c}</math>

The answer is 2.

===Middling===
What is the magnitude of the vector C = A - B if A = <6, 21, 17> and B = <12, 7, 15>?

A - B = <math> <6-12, 21-7, 17-15> = <-6, 14, 2> = C</math>

<math>\sqrt{(-6)^2 + 14^2 + 2^2} = 15.36</math>

===Difficult===
What is the unit vector in the direction of the vector <12, -15, 9>?
First you have to find the magnitude of the vector given:
<math>\sqrt{12^2 + (-15)^2 + 9^2} = 21.21</math>

Finally divide the vector by its magnitude to get the unit vector:

<math>\tfrac{<12,-15,9>}{180}</math>

= <.565, -.707, .424>

==Applications==
You will use vectors for everything in physics ranging from velocity to gravitational field.

==History==

Giusto Bellavitis abstracted the basic idea of a vector in 1835 when he established the concept of equipollence. He called any pair of line segments of the same length and orientation equipollent. He found a relationship and created the first set of vectors.

The name vector was given to us by William Rowan Hamilton as part of his system of quaternions. The vectors he used were three dimensional.

Several other mathematicians developed similar vector systems to those of Bellavitis and Hamilton in the 19th century. The system used by Herman Grassman is the one that is most similar to the one used today. He thought of ideas similar to the cross product and vector differentiation.

== See also ==

===External links===
[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction]

[[Category:Which Category did you place this in?]]

Vectors

2016-04-16T15:46:51Z

Asatapathy3:

'

Written by Elizabeth Robelo
Improved by: Aparajita Satapathy

In physics, a vector is an object with a magnitude and a direction.

==The Main Idea==

A vector is an object with a magnitude and a direction. The magnitude of a vector is a scalar which represents the length of the vector. A vector is represented by an arrow. The orientation of the vector represents its direction. When a vector is drawn, the starting point is the tail and the ending point is called the head of the vector. Refer to the image below for a visual representation:

[[File:http://www.phatcode.net/res/214/images/vector_i.png]]

Vectors can be added and subtracted to each other. To add two vectors you put them head to tail. The connecting arrow starting from the tail of one to the head of the other is the new vector.

[[File:Addingvectors.jpg|275px|thumb|center|Adding vector A to B]]



To subtract two vectors reverse the direction of the one you want to subtract and continue to add them like shown before.



[[File:Subtractingvectors.jpg|350px|thumb|center|Subtracting vector B from A]]

A unit vector is a vector that points in the same direction as the original vector with magnitude 1. We usually designate the unit vector with a "hat" <math alt= r-hat>{\hat{\imath}}</math>. The unit vector is often called the normal vector.

===A Mathematical Model===

Vectors are given by x, y, and z coordinates. They are written in the form <x, y, z> or xi + yj - zk.
Magnitude: <math> |A| = \sqrt{x^2 + y^2 + z^2} </math>

Addition of two vectors: <a1, a2, a3> + <b1, b2, b3> = <a1 + b1, a2 + b2, a3 + b3>

Unit Vector: :<math alt= "u-hat equals the vector u divided by its length">\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}</math>

===A Computational Model===

Here you have vpython code creating a vector from one object to another. [https://trinket.io/embed/glowscript/a8e75ad500 vectors]

==Examples==

===Simple===
Which of the following statements is correct?

[[File:simproblem.jpg|275px|thumb|center]]

1. <math>\overrightarrow{c} = \overrightarrow{b} + \overrightarrow{a}</math>

2. <math>\overrightarrow{a} = \overrightarrow{b} - \overrightarrow{c}</math>

3. <math>\overrightarrow{a} = \overrightarrow{b} + \overrightarrow{c}</math>

4. <math>\overrightarrow{b} = \overrightarrow{a} + \overrightarrow{c}</math>

The answer is 2.

===Middling===
What is the magnitude of the vector C = A - B if A = <6, 21, 17> and B = <12, 7, 15>?

A - B = <math> <6-12, 21-7, 17-15> = <-6, 14, 2> = C</math>

<math>\sqrt{(-6)^2 + 14^2 + 2^2} = 15.36</math>

===Difficult===
What is the unit vector in the direction of the vector <12, -15, 9>?
First you have to find the magnitude of the vector given:
<math>\sqrt{12^2 + (-15)^2 + 9^2} = 21.21</math>

Finally divide the vector by its magnitude to get the unit vector:

<math>\tfrac{<12,-15,9>}{180}</math>

= <.565, -.707, .424>

==Applications==
You will use vectors for everything in physics ranging from velocity to gravitational field.

==History==

Giusto Bellavitis abstracted the basic idea of a vector in 1835 when he established the concept of equipollence. He called any pair of line segments of the same length and orientation equipollent. He found a relationship and created the first set of vectors.

The name vector was given to us by William Rowan Hamilton as part of his system of quaternions. The vectors he used were three dimensional.

Several other mathematicians developed similar vector systems to those of Bellavitis and Hamilton in the 19th century. The system used by Herman Grassman is the one that is most similar to the one used today. He thought of ideas similar to the cross product and vector differentiation.

== See also ==

===External links===
[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction]

[[Category:Which Category did you place this in?]]

Vectors

2016-04-16T15:38:10Z

Asatapathy3:

'

Written by Elizabeth Robelo
Improved by: Aparajita Satapathy

In physics, a vector is an object with a magnitude and a direction.

==The Main Idea==

A vector is an object with a magnitude and a direction. The magnitude of a vector is a scalar. A vector is represented by an arrow. The length of the arrow is the vector’s magnitude and the direction the arrow points is its direction. The start of the arrow is called the tail. The end where the arrow head is located is called the head.

Vectors can be added and subtracted to each other. To add two vectors you put them head to tail. The connecting arrow starting from the tail of one to the head of the other is the new vector.

[[File:Addingvectors.jpg|275px|thumb|center|Adding vector A to B]]



To subtract two vectors reverse the direction of the one you want to subtract and continue to add them like shown before.



[[File:Subtractingvectors.jpg|350px|thumb|center|Subtracting vector B from A]]

A unit vector is a vector that points in the same direction as the original vector with magnitude 1. We usually designate the unit vector with a "hat" <math alt= r-hat>{\hat{\imath}}</math>. The unit vector is often called the normal vector.

===A Mathematical Model===

Vectors are given by x, y, and z coordinates. They are written in the form <x, y, z> or xi + yj - zk.
Magnitude: <math> |A| = \sqrt{x^2 + y^2 + z^2} </math>

Addition of two vectors: <a1, a2, a3> + <b1, b2, b3> = <a1 + b1, a2 + b2, a3 + b3>

Unit Vector: :<math alt= "u-hat equals the vector u divided by its length">\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}</math>

===A Computational Model===

Here you have vpython code creating a vector from one object to another. [https://trinket.io/embed/glowscript/a8e75ad500 vectors]

==Examples==

===Simple===
Which of the following statements is correct?

[[File:simproblem.jpg|275px|thumb|center]]

1. <math>\overrightarrow{c} = \overrightarrow{b} + \overrightarrow{a}</math>

2. <math>\overrightarrow{a} = \overrightarrow{b} - \overrightarrow{c}</math>

3. <math>\overrightarrow{a} = \overrightarrow{b} + \overrightarrow{c}</math>

4. <math>\overrightarrow{b} = \overrightarrow{a} + \overrightarrow{c}</math>

The answer is 2.

===Middling===
What is the magnitude of the vector C = A - B if A = <6, 21, 17> and B = <12, 7, 15>?

A - B = <math> <6-12, 21-7, 17-15> = <-6, 14, 2> = C</math>

<math>\sqrt{(-6)^2 + 14^2 + 2^2} = 15.36</math>

===Difficult===
What is the unit vector in the direction of the vector <12, -15, 9>?
First you have to find the magnitude of the vector given:
<math>\sqrt{12^2 + (-15)^2 + 9^2} = 21.21</math>

Finally divide the vector by its magnitude to get the unit vector:

<math>\tfrac{<12,-15,9>}{180}</math>

= <.565, -.707, .424>

==Applications==
You will use vectors for everything in physics ranging from velocity to gravitational field.

==History==

Giusto Bellavitis abstracted the basic idea of a vector in 1835 when he established the concept of equipollence. He called any pair of line segments of the same length and orientation equipollent. He found a relationship and created the first set of vectors.

The name vector was given to us by William Rowan Hamilton as part of his system of quaternions. The vectors he used were three dimensional.

Several other mathematicians developed similar vector systems to those of Bellavitis and Hamilton in the 19th century. The system used by Herman Grassman is the one that is most similar to the one used today. He thought of ideas similar to the cross product and vector differentiation.

== See also ==

===External links===
[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction]

[[Category:Which Category did you place this in?]]

Vectors

2016-04-16T15:36:12Z

Asatapathy3: /*Applications */

'''CLAIMED BY APARAJITA SATAPATHY'''

Written by Elizabeth Robelo

In physics, a vector is an object with a magnitude and a direction.

==The Main Idea==

A vector is an object with a magnitude and a direction. The magnitude of a vector is a scalar. A vector is represented by an arrow. The length of the arrow is the vector’s magnitude and the direction the arrow points is its direction. The start of the arrow is called the tail. The end where the arrow head is located is called the head.

Vectors can be added and subtracted to each other. To add two vectors you put them head to tail. The connecting arrow starting from the tail of one to the head of the other is the new vector.

[[File:Addingvectors.jpg|275px|thumb|center|Adding vector A to B]]



To subtract two vectors reverse the direction of the one you want to subtract and continue to add them like shown before.



[[File:Subtractingvectors.jpg|350px|thumb|center|Subtracting vector B from A]]

A unit vector is a vector that points in the same direction as the original vector with magnitude 1. We usually designate the unit vector with a "hat" <math alt= r-hat>{\hat{\imath}}</math>. The unit vector is often called the normal vector.

===A Mathematical Model===

Vectors are given by x, y, and z coordinates. They are written in the form <x, y, z> or xi + yj - zk.
Magnitude: <math> |A| = \sqrt{x^2 + y^2 + z^2} </math>

Addition of two vectors: <a1, a2, a3> + <b1, b2, b3> = <a1 + b1, a2 + b2, a3 + b3>

Unit Vector: :<math alt= "u-hat equals the vector u divided by its length">\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}</math>

===A Computational Model===

Here you have vpython code creating a vector from one object to another. [https://trinket.io/embed/glowscript/a8e75ad500 vectors]

==Examples==

===Simple===
Which of the following statements is correct?

[[File:simproblem.jpg|275px|thumb|center]]

1. <math>\overrightarrow{c} = \overrightarrow{b} + \overrightarrow{a}</math>

2. <math>\overrightarrow{a} = \overrightarrow{b} - \overrightarrow{c}</math>

3. <math>\overrightarrow{a} = \overrightarrow{b} + \overrightarrow{c}</math>

4. <math>\overrightarrow{b} = \overrightarrow{a} + \overrightarrow{c}</math>

The answer is 2.

===Middling===
What is the magnitude of the vector C = A - B if A = <6, 21, 17> and B = <12, 7, 15>?

A - B = <math> <6-12, 21-7, 17-15> = <-6, 14, 2> = C</math>

<math>\sqrt{(-6)^2 + 14^2 + 2^2} = 15.36</math>

===Difficult===
What is the unit vector in the direction of the vector <12, -15, 9>?
First you have to find the magnitude of the vector given:
<math>\sqrt{12^2 + (-15)^2 + 9^2} = 21.21</math>

Finally divide the vector by its magnitude to get the unit vector:

<math>\tfrac{<12,-15,9>}{180}</math>

= <.565, -.707, .424>

==Applications==
You will use vectors for everything in physics ranging from velocity to gravitational field.

==History==

Giusto Bellavitis abstracted the basic idea of a vector in 1835 when he established the concept of equipollence. He called any pair of line segments of the same length and orientation equipollent. He found a relationship and created the first set of vectors.

The name vector was given to us by William Rowan Hamilton as part of his system of quaternions. The vectors he used were three dimensional.

Several other mathematicians developed similar vector systems to those of Bellavitis and Hamilton in the 19th century. The system used by Herman Grassman is the one that is most similar to the one used today. He thought of ideas similar to the cross product and vector differentiation.

== See also ==

===External links===
[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction]

[[Category:Which Category did you place this in?]]

Vectors

2016-04-16T15:35:23Z

Asatapathy3:

'''CLAIMED BY APARAJITA SATAPATHY'''

Written by Elizabeth Robelo

In physics, a vector is an object with a magnitude and a direction.

==The Main Idea==

A vector is an object with a magnitude and a direction. The magnitude of a vector is a scalar. A vector is represented by an arrow. The length of the arrow is the vector’s magnitude and the direction the arrow points is its direction. The start of the arrow is called the tail. The end where the arrow head is located is called the head.

Vectors can be added and subtracted to each other. To add two vectors you put them head to tail. The connecting arrow starting from the tail of one to the head of the other is the new vector.

[[File:Addingvectors.jpg|275px|thumb|center|Adding vector A to B]]



To subtract two vectors reverse the direction of the one you want to subtract and continue to add them like shown before.



[[File:Subtractingvectors.jpg|350px|thumb|center|Subtracting vector B from A]]

A unit vector is a vector that points in the same direction as the original vector with magnitude 1. We usually designate the unit vector with a "hat" <math alt= r-hat>{\hat{\imath}}</math>. The unit vector is often called the normal vector.

===A Mathematical Model===

Vectors are given by x, y, and z coordinates. They are written in the form <x, y, z> or xi + yj - zk.
Magnitude: <math> |A| = \sqrt{x^2 + y^2 + z^2} </math>

Addition of two vectors: <a1, a2, a3> + <b1, b2, b3> = <a1 + b1, a2 + b2, a3 + b3>

Unit Vector: :<math alt= "u-hat equals the vector u divided by its length">\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}</math>

===A Computational Model===

Here you have vpython code creating a vector from one object to another. [https://trinket.io/embed/glowscript/a8e75ad500 vectors]

==Examples==

===Simple===
Which of the following statements is correct?

[[File:simproblem.jpg|275px|thumb|center]]

1. <math>\overrightarrow{c} = \overrightarrow{b} + \overrightarrow{a}</math>

2. <math>\overrightarrow{a} = \overrightarrow{b} - \overrightarrow{c}</math>

3. <math>\overrightarrow{a} = \overrightarrow{b} + \overrightarrow{c}</math>

4. <math>\overrightarrow{b} = \overrightarrow{a} + \overrightarrow{c}</math>

The answer is 2.

===Middling===
What is the magnitude of the vector C = A - B if A = <6, 21, 17> and B = <12, 7, 15>?

A - B = <math> <6-12, 21-7, 17-15> = <-6, 14, 2> = C</math>

<math>\sqrt{(-6)^2 + 14^2 + 2^2} = 15.36</math>

===Difficult===
What is the unit vector in the direction of the vector <12, -15, 9>?
First you have to find the magnitude of the vector given:
<math>\sqrt{12^2 + (-15)^2 + 9^2} = 21.21</math>

Finally divide the vector by its magnitude to get the unit vector:

<math>\tfrac{<12,-15,9>}{180}</math>

= <.565, -.707, .424>

==Connectedness==
You will use vectors for everything in physics ranging from velocity to gravitational field.

==History==

Giusto Bellavitis abstracted the basic idea of a vector in 1835 when he established the concept of equipollence. He called any pair of line segments of the same length and orientation equipollent. He found a relationship and created the first set of vectors.

The name vector was given to us by William Rowan Hamilton as part of his system of quaternions. The vectors he used were three dimensional.

Several other mathematicians developed similar vector systems to those of Bellavitis and Hamilton in the 19th century. The system used by Herman Grassman is the one that is most similar to the one used today. He thought of ideas similar to the cross product and vector differentiation.

== See also ==

===External links===
[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction]

[[Category:Which Category did you place this in?]]

Vectors

2016-03-14T16:44:13Z

Asatapathy3:

'''CLAIMED BY APARAJITA SATAPATHY'''

Written by Elizabeth Robelo

A vector is an object with a magnitude and a direction.

==The Main Idea==

A vector is an object with a magnitude and a direction. The magnitude of a vector is a scalar. A vector is represented by an arrow. The length of the arrow is the vector’s magnitude and the direction the arrow points is its direction. The start of the arrow is called the tail. The end where the arrow head is located is called the head.

Vectors can be added and subtracted to each other. To add two vectors you put them head to tail. The connecting arrow starting from the tail of one to the head of the other is the new vector.

[[File:Addingvectors.jpg|275px|thumb|center|Adding vector A to B]]



To subtract two vectors reverse the direction of the one you want to subtract and continue to add them like shown before.



[[File:Subtractingvectors.jpg|350px|thumb|center|Subtracting vector B from A]]

A unit vector is a vector that points in the same direction as the original vector with magnitude 1. We usually designate the unit vector with a "hat" <math alt= r-hat>{\hat{\imath}}</math>. The unit vector is often called the normal vector.

===A Mathematical Model===

Vectors are given by x, y, and z coordinates. They are written in the form <x, y, z> or xi + yj - zk.
Magnitude: <math> |A| = \sqrt{x^2 + y^2 + z^2} </math>

Addition of two vectors: <a1, a2, a3> + <b1, b2, b3> = <a1 + b1, a2 + b2, a3 + b3>

Unit Vector: :<math alt= "u-hat equals the vector u divided by its length">\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}</math>

===A Computational Model===

Here you have vpython code creating a vector from one object to another. [https://trinket.io/embed/glowscript/a8e75ad500 vectors]

==Examples==

===Simple===
Which of the following statements is correct?

[[File:simproblem.jpg|275px|thumb|center]]

1. <math>\overrightarrow{c} = \overrightarrow{b} + \overrightarrow{a}</math>

2. <math>\overrightarrow{a} = \overrightarrow{b} - \overrightarrow{c}</math>

3. <math>\overrightarrow{a} = \overrightarrow{b} + \overrightarrow{c}</math>

4. <math>\overrightarrow{b} = \overrightarrow{a} + \overrightarrow{c}</math>

The answer is 2.

===Middling===
What is the magnitude of the vector C = A - B if A = <6, 21, 17> and B = <12, 7, 15>?

A - B = <math> <6-12, 21-7, 17-15> = <-6, 14, 2> = C</math>

<math>\sqrt{(-6)^2 + 14^2 + 2^2} = 15.36</math>

===Difficult===
What is the unit vector in the direction of the vector <12, -15, 9>?
First you have to find the magnitude of the vector given:
<math>\sqrt{12^2 + (-15)^2 + 9^2} = 21.21</math>

Finally divide the vector by its magnitude to get the unit vector:

<math>\tfrac{<12,-15,9>}{180}</math>

= <.565, -.707, .424>

==Connectedness==
You will use vectors for everything in physics ranging from velocity to gravitational field.

==History==

Giusto Bellavitis abstracted the basic idea of a vector in 1835 when he established the concept of equipollence. He called any pair of line segments of the same length and orientation equipollent. He found a relationship and created the first set of vectors.

The name vector was given to us by William Rowan Hamilton as part of his system of quaternions. The vectors he used were three dimensional.

Several other mathematicians developed similar vector systems to those of Bellavitis and Hamilton in the 19th century. The system used by Herman Grassman is the one that is most similar to the one used today. He thought of ideas similar to the cross product and vector differentiation.

== See also ==

===External links===
[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction https://en.wikipedia.org/wiki/Euclidean_vector#Addition_and_subtraction]

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Vectors

2016-03-13T14:21:25Z

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Vectors

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