CHAPTER 6 6-4 Vectors and dot products

OBJECTIVES

Find the dot product of two vectors and use the properties

Find the angle between two vectors Write vectors as the sum of two vectors

components Use vectors to find the work done by a force

DOT PRODUCT OF TWO VECTORS

Dot Product is a third vector operation. This vector operation yields a scalar (a single number) not another vector. The dot product can be positive, zero or negative.

Definition of dot product

2121 bbaa wvjiwjiv 2211 and where baba

EXAMPLE

find ,4 and 52 If wvjiwjiv

1542 wv 8 5 3

This is called the dot product. Notice the answer is just a number NOT a vector.

EXAMPLE #1: FINDING DOT PRODUCTS

Find each dot product.

A) <4, 5>●<2, 3> sol: 23

B) <2, -1>●<1, 2> sol:0

C) <0, 3>●<4, -2> sol:-6

D) <6, 3>●<2, -4> sol: 0

E) (5i + j)●(3i – j) sol:14

PROPERTIES OF DOT PRODUCTS

Let u, v, and w be vectors in the plane or in space and let c be a scalar.1. u●v = v●u2. 0●v = 03. u●(v + w) = u●v + u●w4. v●v = ||v||2

5. c(u●v) = cu●v = u●cv

EXAMPLE#2 USING PROPERTIES OF DOT PRODUCTS

Let u=<-1,3>, v=<2,-4> and w=<1,-2>. Use vectors and their properties to find the indicated quantity

A. (u.v)w sol: <-14,28> B.u.2v sol: -28 C.||u|| sol:

CHECK IT OUT!

Let u = <1, 2>, v = <-2, 4> and w = <-1, -2>. Find the dot product.

A) (u●v)w

B) u●2v

CHECK IT OUT !

Given the vectors u = 8i + 8j and v = —10i + 11j find the following.

A. B.

vuvv

ANGLE BETWEEN TWO VECTORS

Angle between two vectors (θ is the smallest non-negative angle between the two vectors)

wvwv

cos

wvwv1cos

EXAMPLE

Find the angle between

2,3,5,1 vu

SOLUTION

2222 23,)5(1

132531

2,3,5,1

vu

vu

vu

vu

vuCos

SOLUTION

457071.0

...7071.0

338

13

1326

13

23,)5(1

132531

2,3,5,1

1

2222

Cos

Cos

Cos

vu

vu

vu

CHECK IT OUT!

Find the angle between u = <4, 3> and v = <3, 5>.

Solution: 22.2 degrees

DEFINITIONS OF ORTHOGONAL VECTORS

The vectors u and v are orthogonal if u●v = 0.

Orthogonal = Perpendicular = Meeting at 90°

EXAMPLE: ORTHOGONAL VECTORS

Are the vectors u = <2, -3> and v = <6, 4> orthogonal?

EXAMPLE

Determine if the pair of vectors is orthogonal

jivjiu316

2and38

jivjiu 921and37

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 Find the projection of u into v. Then write u

as the sum of two orthogonal vectors Sol:

vv

vuuprojw

wwu

vu

v

21

21

2,6,5,3

SOLUTION

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

WORK

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

ABW FThis means the force is in some direction given by the vector F but the line of motion of the object is along a vector from A to B

DEFINITION OF WORK

Work is force times distance.If Force is a constant and not at an angle

If Force is at an angle

PQFprojWPQ

EXAMPLE

To close a barn’s sliding door, a person pulls on a rope with a constant force of 50 lbs. at a constant angle of 60 degrees. Find the work done in moving the door 12 feet to its closed position.

Sol: 300 lbs

EXAMPLE

Find the work done by a force of 50 kilograms acting in the direction 3i + j in moving an object 20 metres from (0, 0) to (20, 0).

SOLUTION

Let's find a unit vector in the direction 3i + j

10133 22 ji

jiu10

1

10

3

Our force vector is in this direction but has a magnitude of 50 so we'll multiply our unit vector by 50.

jiF10

1

10

350

SOLUTION

jiji 02010

1

10

350

W

kgs m 948.68 10

3000W

STUDENT GUIDED PRACTICE

Do7,9,11,23,31,43 and 57 in your book page 440 and 441

HOMEWORK

Do 8,10,12,14,24,32,44 and 69 in your book page 440 and 441

CLOSURE

Today we learned about vectors and work Next class we are going to learn about

trigonometric form of a complex number