Section 13.3 The Dot Product

We have added and subtracted vectors, what about multiplying vectors? There are two ways we can multiply vectors 1.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

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

Geometric Definition of Dot Product Let then

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

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

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 = (x 0, y 0, z 0 ) Then what can we say?

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

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

So we have that Also Computing a vector projection

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!