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VG.3 The Dot Product

So far, we have added two vectors and multiplied a vector by a scalar, but what if we want to multiply two vectors so that the product is a useful quantity? There are two ways to find the product of two vectors, one is the dot product and the other is the cross product. This section will focus on the dot product.

The Dot Product

If and , then the dot product of a and b is the number $\QTR{bf}{a\cdot b}$ given by

Or in two-dimensions,

Notice that the result is a scalar, not a vector.

Properties of the Dot Product

If a, b and c are vectors and c is a scalar then

Another way to define the dot product is to describe it using the angle between vectors a and b.

13.3DotProduct__13.png

Theorem

If $\theta$ is the angle between the vectors a and b, then

The previous theorem can be rearranged to form the following

Corollary

If $\theta$ is the angle between the vectors a and b, then

MATH

Something else that the dot product can do is determine if two vectors are orthogonal (which is a fancy word that means the angle between the two vectors is 90 $\U{b0}$ )

Direction Angles and Direction Cosines

The direction angles of a non-zero vector a are the angles $\alpha ,\beta ,$ and $\gamma$ in the interval that a makes with the positive x-,y-, and z- axes. The cosine of these direction angles are called the direction cosines.

13.3DotProduct__26.png

Using the corollary introduced in this chapter, we replace the second vector, b, with unit vector i.

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Since i equals one, it can we call let it disappear (remembering that it is the vector that a is being compared to)

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Using the other two unit vectors, we can make the following two equations

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From here we can see the direction cosines, it is simply the result of the right side of the equation. Similarly, when you solve for $\alpha ,\beta ,$ and $\gamma$ ; you will have the direction angles.

Projections

Two vectors a and b, have the same origin. If one were to make a line perpendicular to vector a and connect it with the end of the vector b, the vector formed by the origin and point p would be considered the vector projection of b onto a, or proj $_{\QTR{bf}{a}}$ b. The scalar projection of b onto a (or comp $_{\QTR{bf}{a}}$ b) is the magnitude of the vector projection.

Projections

Scalar projection of b onto a:

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Vector projection of b onto a:

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Notice that the projections are of b onto a. If one wants the projection of a onto b it would look like this

Vector projection of a onto b:

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Also notice that the vector projection is the scalar projection times the unit vector in the direction of a.

13.3DotProduct__49.png

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