# 10.2 - Vectors

- StudyBlue
- North-carolina
- University of North Carolina - Charlotte
- Mathematics
- Mathematics 2241
- Gordon
- 10.2 - Vectors

**Created:**2016-12-20

**Last Modified:**2016-12-27

**Definition of Vector Addition**

**u**and

**v**are vectors positioned so the initial point of

**v**is at the terminal point of

**u**, then the sum: ______ is the vector from the initial point of u to the terminal point of

**v**.

**• u - v**

**• u + v**

**u**+

**v**

**Definition of Scalar Multiplication:**

*c*is a scalar and

**v**is a vector, then the scalar multiple

*c*

**v**is the vector whose length is |

*c*| times the length of

**v**and whose direction is the same as v if

**1) ____**and is opposite to

**v**if

**2) ____**.

*c*> 0) or (

*c*< 0) or (

*c*= 0)

*c*> 0) or (

*c*< 0) or (

*c*= 0)

*c*> 0

*c*< 0

**Vector Representation**: Given the points

*A(x*&

_{1},y_{1},z_{1})*B(x*, the vector

_{2},y_{2},z_{2})**a**with representation

*A*is:

^{→}B**a**= (x

_{2}+ x

_{1}, y

_{2}+ y

_{1}, z

_{2}+ z

_{1})

**a**=〈x

_{2}− x

_{1}, y

_{2}− y

_{1}, z

_{2}− z

_{1}〉

**a**=〈x

_{2}− x

_{1}, y

_{2}− y

_{1}, z

_{2}− z

_{1}〉

__length__of the 2-D vector

**a**= 〈 a

_{1}, a

_{2}〉is

- |
**a**| = √(a_{1}^{2}+ a_{2}^{2}) - |
**a**| = √(a_{1}^{2}+ a_{2}^{2}+ a_{3}^{2})

**a**| = √(a

_{1}

^{2}+ a

_{2}

^{2})

__length__of the 3-D vector

**a**=〈 a

_{1}, a

_{2}, a

_{3}〉is:

- |
**a**| = √(a_{1}^{2}+ a_{2}^{2}+ a_{3}^{2}) - |
**a**| = √(a_{1}^{2}+ a_{2}^{2})

**a**| = √(a

_{1}

^{2}+ a

_{2}

^{2}+ a

_{3}

^{2})

**a**=〈a

_{1},a

_{2}〉then:

*c***a**=〈ca_{1}, ca_{2}〉*c***a**=〈ca_{1}+ ca_{2}〉

*c*

**a**=〈ca

_{1}, ca

_{2}〉

**a**=〈a

_{1},a

_{2}〉then:

**a**+

**b**=

- 〈a
_{1}− b_{1}, a_{2}− b_{2}〉 - 〈a
_{1}+ b_{1}, a_{2}+ b_{2}〉

_{1}+ b

_{1}, a

_{2}+ b

_{2}〉

**a**=〈a

_{1},a

_{2}〉then:

**a**−

**b**=

- 〈a
_{1}− b_{1}, a_{2}− b_{2}〉 - 〈a
_{1}+ b_{1}, a_{2}+ b_{2}〉

_{1}− b

_{1}, a

_{2}− b

_{2}〉

_{1},a

_{2},a

_{3}〉+〈b

_{1},b

_{2},b

_{3}〉=

- 〈a
_{1}+b_{1}, a_{2}+b_{2}, a_{3}+b_{3}〉 - 〈a
_{1}−b_{1}, a_{2}−b_{2}, a_{3}−b_{3}〉

_{1}+b

_{1}, a

_{2}+b

_{2}, a

_{3}+b

_{3}〉

_{1},a

_{2},a

_{3}〉−〈b

_{1},b

_{2},b

_{3}〉=

- 〈a
_{1}+b_{1}, a_{2}+b_{2}, a_{3}+b_{3}〉 - 〈a
_{1}−b_{1}, a_{2}−b_{2}, a_{3}−b_{3}〉

_{1}−b

_{1}, a

_{2}−b

_{2}, a

_{3}−b

_{3}〉

**Properties Of Vectors**: If

**a**,

**b**, and

**c**are vectors in

*V*and

_{n}*c*and

*d*are scalars, then:

**a + b**=

**b + a***c***a +***c***b**

**b**+

**a**

**Properties Of Vectors**: If

**a**,

**b**, and

**c**are vectors in

*V*and

_{n}*c*and

*d*are scalars, then:

**a + 0**=

**a + b****a**

**a**

**Properties Of Vectors**: If

**a**,

**b**, and

**c**are vectors in

*V*and

_{n}*c*and

*d*are scalars, then:

*c*(

**a + b**) =

**a + b***c***a**+*c***b**

*c*

**a**+

*c*

**b**

**Properties Of Vectors**: If

**a**,

**b**, and

**c**are vectors in

*V*and

_{n}*c*and

*d*are scalars, then: (

*cd*)

**a**=

**a***c*(*d***a**)*c*

*d*

**a**)

**Properties Of Vectors**: If

**a**,

**b**, and

**c**are vectors in

*V*and

_{n}*c*and

*d*are scalars, then:

**a**+ (

**b**+

**c**) =

- (
**a**+**b**) +**c** **ab**+**ac**

**a**+

**b**) +

**c**

**Properties Of Vectors**: If

**a**,

**b**, and

**c**are vectors in

*V*and

_{n}*c*and

*d*are scalars, then:

**a**+ (−

**a**) =

**a****0**

**0**

**Properties Of Vectors**: If

**a**,

**b**, and

**c**are vectors in V

_{n}and c and d are scalars, then: (

*c*+

*d*)

**a**=

*c***a**+*c***b***c***a**+*d***a**

*c*

**a**+

*d*

**a**

**Properties Of Vectors**: If

**a**,

**b**, and

**c**are vectors in V

_{n}and c and d are scalars, then: 1

**a**=

**a****b**+**a**

**unit vector**is a vector whose length is 1. In general, if

**a ≠ 0**, then the unit vector that has the same direction as

**a**is:

**u**= 1 / |**a**|**u**= ( 1 / |**a**| ) (**a**) =**a**/ |**a**|

**u**= ( 1 / |

**a**| ) (

**a**) =

**a**/ |

**a**| = 1

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