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Vectors

Dr .Hayk MelikyanDepartmen of Mathematics and CS

melikyan@nccu.edu

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A line segment to which a direction has been assigned is called a directed line segment. The figure below shows a directed line segment form P to Q. We call P the initial point and Q the terminal point. We denote this directed line segment by PQ.

The magnitude of the directed line segment PQ is its length. We denote this by || PQ ||. Thus, || PQ || is the distance from point P to point Q. Because distance is nonnegative, vectors do not have negative magnitudes.

Geometrically, a vector is a directed line segment. Vectors are often denoted by a boldface letter, such as v. If a vector v has the same magnitude and the same direction as the directed line segment PQ, we write

v = PQ.

P

Q

Initial point

Terminal point

Directed Line Segments and Geometric Vectors

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Vector Multiplication

If k is a real number and v a vector, the vector kv is called a scalar multiple of the vector v. The magnitude and direction of kv are given as follows:

The vector kv has a magnitude of |k| ||v||. We describe this as the absolute value of k times the magnitude of vector v.

The vector kv has a direction that is: the same as the direction of v if k > 0, and opposite the direction of v if k < 0

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A geometric method for adding two vectors is shown below. The sum of u + v is called the resultant vector. Here is how we find this vector.

1. Position u and v so the terminal point of u extends from the initial point of v.

2. The resultant vector, u + v, extends from the initial point of u to the terminal point of v.

Initial point of u

u + vv

u

Resultant vector

Terminal point of v

The Geometric Method for Adding Two Vectors

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The difference of two vectors, v – u, is defined as v – u = v + (-u), where –u is the scalar multiplication of u and –1: -1u. The difference v – u is shown below geometrically.

v

u-u

-u

v – u

The Geometric Method for the Difference of Two Vectors

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1

1

i

j

Ox

y

The i and j Unit Vectors

Vector i is the unit vector whose direction is along the positive x-axis. Vector j is the unit vector whose direction is along the positive y-axis.

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Representing Vectors in Rectangular CoordinatesVector v, from (0, 0) to (a, b), is represented as v = ai + bj.The real numbers a and b are called the scalarcomponents of v. Note that a is the horizontalcomponent of v, and b is the vertical component of v.The vector sum ai + bj is called a linear combinationof the vectors i and j. The magnitude of v = ai + bj isgiven by

v a2 b2

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Sketch the vector v = -3i + 4j and find its magnitude.

Solution For the given vector v = -3i + 4j, a = -3 and b = 4. The vector, shown below, has the origin, (0, 0), for its initial point and (a, b) = (-3, 4) for its terminal point. We sketch the vector by drawing an arrow from (0, 0) to (-3, 4). We determine the magnitude of the vector by using the distance formula. Thus, the magnitude is

-5 -4 -3 -2 -1 1 2 3 4 5

5

4

3

2

1

-1-2

-3

-4-5

Initial point

Terminal point

v = -3i + 4j

v a2 b2

( 3)2 42

9 16

25 5

Text Example

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Representing Vectors in Rectangular Coordinates

• Vector v with initial point P1 = (x1, y1) and terminal point

P2 = (x2, y2) is equal to the position vector

v = (x2 – x1)i + (y2 – y1)j.

Adding and Subtracting Vectors in Terms of i and j

If v = a1i + b1j and w = a2i + b2j, then v + w = (a1 + a2)i + (b1 + b2)j

v – w = (a1 – a2)i + (b1 – b2)j

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If v = 5i + 4j and w = 6i – 9j, find: a. v + w b. v – w.

Solution • v + w = (5i + 4j) + (6i – 9j) These are the given vectors.

= (5 + 6)i + [4 + (-9)]j Add the horizontal components. Add the vertical components.

= 11i – 5j Simplify.

• v + w = (5i + 4j) – (6i – 9j) These are the given vectors.= (5 – 6)i + [4 – (-9)]j Subtract the horizontal components.

Subtract the vertical components.= -i + 13j Simplify.

Text Example

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Scalar Multiplication with a Vector in Terms of i and j

• If v = ai + bj and k is a real number, then the scalar multiplication of the vector v and the scalar k is

• kv = (ka)i + (kb)j.

Example: If v = 2i - 3j, find 5v and -3v

ji

jiv

ji

jiv

96

)3*3()2*3(3

1510

)3*5()2*5(5

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The Zero VectorThe vector whose magnitude is 0 is called the zero vector, 0. Thezero vector is assigned no direction. It can be expressed in terms of Iand j using• 0 = 0i + 0j.

Properties of Vector Addition

If u, v, and w are vectors, then the following properties are true.

Vector Addition Properties 1. u + v = v + u Commutative Property 2. (u + v) + w = v + (u + w) Associative Property 3. u + 0 = 0 + u = u Additive Identity 4. u + (-u) = (-u) + u = 0 Additive Inverse

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Properties of Vector Addition and Scalar MultiplicationIf u, v, and w are vectors, and c and d are scalars, then the followingproperties are true.

Scalar Multiplication Properties 1. (cd)u = c(du) Associative Property 2. c(u + v) = cu + cv Distributive Property 3. (c + d)u = cu + du Distributive Property 4. 1u = u Multiplicative Identity 5. 0u = 0 Multiplication Property 6. ||cv|| = |c| ||v||

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Finding the Unit Vector that Has the Same Direction as a Given Nonzero Vector v

For any nonzero vector v, the vector

is a unit vector that has the same direction as v. To find this

vector, divide v by its magnitude.

v

v

Example

Find a unit vector in the same direction as v=4i-7j

jiv

v

v

65

7

65

4

654916

)7(4 22

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Definition of a Dot Product

If v=a1i+b1j and w = a2i+b2j are vectors, the dot product is defined as

2121 bbaawv The dot product of two vectors is the sum of the products of their horizontal and vertical components.

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If v = 5i – 2j and w = -3i + 4j, find: a. v · w b. w · v c. v · v.

Solution To find each dot product, multiply the two horizontal components, and then multiply the two vertical components. Finally, add the two products. a. v · w = 5(-3) + (-2)(4) = -15 – 8 = -23

b. w · v = (-3)(5) + (4)(-2) = -15 – 8 = -23

c. v · v = (5)(5) + (-2)(-2) = 25 + 4 = 29

Text Example

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Properties of the Dot Product

If u, v, and w, are vectors, and c is a scalar, then 1. u · v = v · u 2. u · (v + w) = u · v + u · w 3. 0 · v = 0 4. v · v = || v ||2

5. (cu) · v = c(u · v) = u · (cv)

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Alternative Formula for the Dot Product• If v and w are two nonzero vectors and is

the smallest nonnegative angle between them, then

v · w = ||v|| ||w|| cos.

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Formula for the Angle between Two Vectors

wv

wvand

wv

wv

1coscos

If v and w are two nonzero vectors and is thesmallest nonnegative angle between v and w,

then

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Example

Find the angle between v=2i-4j and w=3i+2j.

Solution:

1.9765

1cos

652

2cos

1320

86cos

23*)4(2

2*43*2cos

cos

1

11

2222

1

1

wv

wv

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The Dot Product and Orthogonal Vectors

Two nonzero vectors v and w are orthogonal if and only if v•w=o.

Because 0•v=0, the zero vector is orthogonal to every vector v.

Example

Are the vectors v=3i-2j and w=3i+2j orthogonal?

0549

2*23*3

wv

The vectors are not orthogonal.

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The Vector Projection of v Onto w

If v and w are two nonzero vectors, the vector projection of v onto w is

ww

wvvprojw 2

If v=3i+4j and w=2i-5j, find the projection of v onto wSolution:

jiji

ji

ww

wvvprojw

29

70

29

28)52(

29

14

)52(])5(2[

5*42*322

2

Example

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The Vector Components of v

Let v and w be two nonzero vectors. Vector v can be expressedas the sum of two orthogonal vectors v1 and v2, where v1 is

parallel to w and v2 is orthogonal to w.

1221 , vvvww

wvvprojv w

Thus, v = v1 + v2. The vectors v1 and v2 are called the vector components of v. The process of expressing v as v1 and v2 is called the decomposition of v into v1 and v2.

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Example

Let v=3i+j and w=2i-3j. Decompose v into two vectors, where one is

parallel to w and the other is orthogonal to w.

Solution:

ji

jiv

jiji

jiv

vvvww

wvv

13

30

13

33

)13

93()

13

63(

13

9

13

6)32(

13

3

)32()3(2

3*12*3

,

2

221

1221

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Definition of Work

• The work W done by a force F in moving an object from A to B is

• W = F · AB.