Vector Dot Products

Introduction: In this lesson we will examine a combination of vectors known as the dot product. Vector components will be combined in such a way as to result in a scalar (number). Applications of the dot product will be shown.

In general, if v = (v₁, v₂) and u = (u₁, u₂), the dot product .

In three dimensions if v = (v₁, v₂, v₃) and u = (u₁, u₂, u₃), the dot product .

Work, W , is the product of the force and the distance through which the force is applied. It can be represented by a dot product:

where F is the applied force which may or may not be entirely in the same direction as s, the distance the object moves.

Let v = (2, 5) and u = (–3, 2) be two 2 dimensional vectors. The dot product of v and u would be given by .

A dot product can be used to calculate the angle between two vectors. Suppose that v = (5, 2) and u = (–3, 1) as shown in the diagram shown below. We wish to calculate angle

between v and u.

To do this we use the formula

which can be derived using the Law of Cosines and the fact that .

This gives us

allowing us to calculate the angle .

Generalizing, we can calculate the angle between any two vectors u and v by using the dot product of the unit vectors in the same direction as v and u in this formula

1. A constant force of 50 pounds is applied at an angle of 60º to pull a 12-foot sliding metal door shut. The diagram shown below illustrates this situation.

F is the applied force and s is the vector representing the direction the door slides.

We can represent these vectors as s = (12, 0) and F = .

Simplifying F yields .

We can now form the dot product and get our asnwer:

foot-pounds.

Notice that only the horizontal component of F affects the work. This result can also be found using the formula

.

2. What is the angle between i = (1, 0) and j = (0, 1)?

We choose this example because we know that the angle between these basic unit vectors is 90º.

Verifying this information with our formulas yields:

and