Section 13.3The Dot Product

• We have added and subtracted vectors, what about multiplying vectors?

• There are two ways we can multiply vectors1. One results in a scalar product and is called the

dot product

2. The other results in a vector product and is called the cross product

• We will focus today on the first of these two products

– The other we will cover next section

• Consider a child pulling a wagon along a horizontal sidewalk– Assume the child pulls with 2 lbs of force and

moves the wagon 1 ft, then the amount of work done will depend the angle of the handle with respect to the ground

1F

2F3F

• Assume that the force vector is lbs and the distance vector is feet– What is the work done?– What if the force vector is– What if the force vector is – Note in each case the magnitude of the force is 2

1F

2F3F

i

2i

1

ji

22 ji

3

Geometric Definition of Dot Product• Let

then jwiwwwww

jvivvvvv

2121

2121

or,

andor,

0andandbetween

angletheiswherecos

wv

wvwv

Algebraic Definition of Dot Product• Let

then

• Notice in both cases our result is a number (or a scalar)

• We have given these definitions in 2 dimensions, they hold for n dimensions

• Let’s see why they both work

jwiwwwww

jvivvvvv

2121

2121

or,

andor,

2211 wvwvwv

Some properties of the dot product

• If two nonzero vectors and are perpendicular, or orthogonal, then

• The magnitude and dot product are related as well

uwuvuwv

wvwvwv

vwwv

wvu

)(.3

)()()(.2

1.

scalarabeandvectorsbeand,,Let

0wv v

w

2v v v

Normal Vectors• A normal vector to a plane is a vector that is

perpendicular to the plane

• Suppose we have a plane, Π, which passes thru P = (x, y, z) and has a normal vector

– We will start by identifying another point in the plane, say Q = (x0, y0, z0)

• Then what can we say?

cban ,,

• Then we have

• This is called point normal form for an equation of a plane

• Notice all we need is a point in the plane and a vector normal to the plane to find its equation

0)()()(

0,,,,

0so

000

000

zzcyybxxa

zzyyxxcba

nPQnPQ

Vector Projections

• We may be interested in what influence one vector has on another– For example, we may wonder how much force

wind exerts on an airplane trajectory (when they are not going the same direction)

• In this case we are looking for the vector projection of the wind in the direction of the airplane vector

Computing a vector projection

• We are looking for the vector projection of in the direction of (which is a unit vector)

• Then

v

u

uv

parallelvv

)1since(cos uuvvvparallel

• So we have that

• Also

Computing a vector projection

uuvuvvparallel

)()cos(

parallelparallel vvvvvv

so

• Back to our discussion of work from the beginning

• In physics, work done by a force of magnitude F that moves an object a distance d is given by W = Fd, provided the force and the displacement are in the same direction

• Thus to calculate the work done, we must make sure that we compute the component of force that is in the same direction that the object is being moved

• We need magnitude of the vector projection of the force in the direction of movement which is just the dot product of the two!