Page 235 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

(b) , ,

Prove: If is the angle between u and v and , then .
20. .

Consider the parallelepiped with sides , , and
21.

(a) Find the area of the face determined by u and w.

(b) Find the angle between u and the plane containing the face determined by v and w.

Note The angle between a vector and a plane is defined to be the complement of the angle θ between the vector and

that normal to the plane for which .

Find a vector n that is perpendicular to the plane determined by the points , , and .
22. [See the note in Exercise 21.]

Let m and n be vectors whose components in the -system of Figure 3.4.10 are and .
23.

(a) Find the components of m and n in the -system of Figure 3.4.10.

(b) Compute using the components in the -system.

(c) Compute using the components in the -system.

(d) Show that the vectors obtained in (b) and (c) are the same.

Prove the following identities.
24.

(a)

(b)

Let u, v, and w be nonzero vectors in 3-space with the same initial point, but such that no two of them are collinear. Show that
25.

(a) lies in the plane determined by v and w

(b) lies in the plane determined by u and v

Prove part (d) of Theorem 3.4.1.
26.