6.4 Vectors and Dot Products

Finding the angle between two vectors

Writing a vector as the sum of two vectors components

Definition of Dot Product

Given: Two vectors in Component form

The result is a number, not a vector

2211

2121 ,,

vuvuvu

vvvanduuu

Find the Dot Product

Given 3,26,4

Find the Dot Product

Given

26188

3624

3,26,4

Products of the Dot Product

cvuorvcuvuc

scalariscwhere

vvv

wuvuwvu

v

uvvu

2

00

The Angle Between two vectors

For angles 0

vu

vuCos

The Angle Between two vectors

For angles

Find the angle between

0

vu

vuCos

2,3,5,1 vu

2222 23,)5(1

132531

2,3,5,1

vu

vu

vuvu

vuCos

457071.0

...7071.0

338

13

1326

13

23,)5(1

132531

2,3,5,1

1

2222

Cos

Cos

Cos

vu

vu

vu

vu

vuCos

The Angle Between two vectors

For angles

Vectors are Orthogonal if there Dot Product (u●v)= 0What is the angle between the vectors,

Why?

0

vu

vuCos

Definition of Vector Components

Let u and v be nonzero vectors.

u = w1 + w2 and w1 · w2 = 0

Also, w1 is a scalar of v

The vector w1 is the projection of u onto v, So w1 = proj v u

w 2 = u – w 1

vv

vuuprojv

2

Decomposing of a Vector Using Vector Components

vv

vuuprojw

wwu

vu

v

21

21

2,6,5,3

Decomposing of a Vector Using Vector Components

5

2,5

62,6

40

8

2,626

2)5(63

2,6,5,3

1

222

1

21

21

w

w

vv

vuuprojw

wwu

vu

v

Decomposing of a Vector Using Vector Components

5

2,5

62,6

40

8

2,626

2)5(63

2,6,5,3

1

222

1

21

21

w

w

vv

vuuprojw

wwu

vu

v

5

27,5

9

5

2,5

6

5

27,5

9

5

25,

5

63

2

2

u

w

w

Definition of Work

Work is force times distance.

If Force is a constant and not at an angle

If Force is at an angle

PQFW

PQFprojWPQ

Homework

Page 447 – 448

# 1, 5, 17, 21, 25,

29, 33, 37, 41,

45, 49, 53, 63, 67

Homework

Page 447 – 448

# 4, 8, 12, 16,

20, 24, 28, 32,

36, 40, 52