Page 210 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Figure 3.3.4
The following theorem shows how the dot product can be used to obtain information about the angle between two vectors; it
also establishes an important relationship between the norm and the dot product.
THEOREM 3.3.1

Let u and v be vectors in 2- or 3-space.
(a) ; that is,

(b) If the vectors u and v are nonzero and θ is the angle between them, then

Proof (a) Since the angle θ between v and v is 0, we have

Proof (b) Since θ satisfies , it follows that θ is acute if and only if , that θ is obtuse if and only if ,
.
and that if and only if . But has the same sign as since , , and

Thus, the result follows.

EXAMPLE 4 Finding Dot Products from Components
If , , and , then