**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.

**Definitions:** In general, if **v** = (v_{1}, v_{2}) and **u** = (u_{1}, u_{2}), the __dot product__.

In three dimensions if **v** = (v_{1}, v_{2}, v_{3}) and **u** = (u_{1}, u_{2}, u_{3}), 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.

**The Lesson:** 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

**Let's Practice:**- 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 .

- 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