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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.

Definitions:
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.
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:
  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
.
  1. 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

Examples
Example
What is the angle between v = (3, –7) and u = (–1, 9)?
What is your answer?
 
Example A force of 20 Newtons is applied to an object at an angle of -45º with the horizontal. The object is pulled 10 meters at an angle of 25º with the horizontal. How much work is done while moving the object?
What is your answer?
 



M Ransom

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