Page 207 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

3.3 In this section we shall discuss an important way of multiplying vectors in
2-space or 3-space. We shall then give some applications of this
DOT PRODUCT; multiplication to geometry.
PROJECTIONS

Dot Product of Vectors

Let u and v be two nonzero vectors in 2-space or 3-space, and assume these vectors have been positioned so that their initial

points coincide. By the angle between u and v, we shall mean the angle θ determined by u and v that satisfies (Figure

3.3.1).

Figure 3.3.1 .
The angle θ between u and v satisfies

DEFINITION

If u and v are vectors in 2-space or 3-space and θ is the angle between u and v, then the dot product or Euclidean inner
product is defined by

(1)

EXAMPLE 1 Dot Product
As shown in Figure 3.3.2, the angle between the vectors and is 45°. Thus