Page 209 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Finding the Angle Between Vectors (4)
(5)
If u and v are nonzero vectors, then Formula 1 can be written as

EXAMPLE 2 Dot Product Using (3)

Consider the vectors and . Find and determine the angle θ between u and v.

Solution

For the given vectors we have , so from (5),

Thus, .

EXAMPLE 3 A Geometric Problem ,
Find the angle between a diagonal of a cube and one of its edges.

Solution

Let k be the length of an edge and introduce a coordinate system as shown in Figure 3.3.4. If we let
, and , then the vector

is a diagonal of the cube. The angle θ between d and the edge satisfies

Thus

Note that this is independent of k, as expected.