11.3 The Dot Product of Two Vectors

1 2 1 2Let , , , .u u v v u v

1 1 2 2u v u v u v

1 1 2 2 3 3u v u v u v u v

The dot product of u and v in the plane is

The dot product of u and v in space is

Two vectors u and v are orthogonal if they meet at a right angle. if and only if u ∙ v = 0 (since slopes are opposite reciprocal)

(Read “u dot v”)

Definitions

3,4 5,2 3 5 4 2 23

2, 3 3,2 2 3 3 2 0

Examples

Let , , be vectorsu v w

2

1.

2. ( )

3. ( )

4. 0

5.

c c c

u v v u

u v w u v u w

u v u v u v

0 v

v v v

cos u v u vAnother form of the Dot Product:

where is the angle between two nonzero vectors and . u v

Properties

Find the angle between vectors u and v:

2,3 , 2,5 u v

1cos

u v

u v1 11

cos13 29

55.5

Examples

Angles between a vector v and 3 unit vectors i, j and k are called direction angles of v, denoted by α, β, and γ respectively. Since

1 2 3 1 cos , , 1,0,0v v v v v i v

31 2cos cos cosvv v

v v v

we obtain the following 3 direction cosines of v:

So any vector v has the normalized form:

cos cos cos v

i j kv

Direction Cosines

1 2v

u vproj v

v

2 1

w u

w = u - w

Let u and v be nonzero vectors. w1 is called the vector component of u along v

(or projection of u onto v), and is denoted by projvu w2 is called the vector component of u orthogonal to v

w2 w1

u

v

Vector Components

A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they?

N

E

Application

A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they?

N

Eu

A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they?

N

E

v

u

60o

A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they?

N

E

v

u

We need to find the magnitude and direction of the resultant vector u + v.

u+v

N

E

v

u

The component forms of u and v are:

u+v

500,0u

70cos60 ,70sin 60v

500

70

35,35 3v

Therefore: 535,35 3 u v

538.4 22535 35 3 u v

and: 1 35 3tan

535 6.5

N

E

The new ground speed of the airplane is about 538.4 mph, and its new direction is about 6.5o north of east.

538.4

6.5o

1) Compute

1,7, 2 , 2, 2,6 , 1, 1,3 u v w

( ) u v w

u v u w2) Compute

4) Find the angle between vectors v and w.

3) List pairs of orthogonal and/or parallel vectors.

6) Find the projection of w onto u.

5) Find the unit vector in the direction u.

7) Find vector component of w orthogonal to u.

Examples