Page 230 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Figure 3.4.8 that the area A of the parallelogram determined by u and v is
It now follows from Theorem 3.4.3 and the fact that

which completes the proof.

Proof (b) As shown in Figure 3.4.8b, take the base of the parallelepiped determined by u, v, and w to be the parallelogram

determined by v and w. It follows from Theorem 3.4.3 that the area of the base is and, as illustrated in Figure 3.4.8b, the

height h of the parallelepiped is the length of the orthogonal projection of u on . Therefore, by Formula 10 of Section 3.3,

It follows that the volume V of the parallelepiped is
so from 7,

which completes the proof.

Remark If V denotes the volume of the parallelepiped determined by vectors u, v, and w, then it follows from Theorem 3.3 and
Formula 7 that

(8)

From this and Theorem 3.3.1 b, we can conclude that

where the + or − results depending on whether u makes an acute or an obtuse angle with .

Formula 8 leads to a useful test for ascertaining whether three given vectors lie in the same plane. Since three vectors not in the

same plane determine a parallelepiped of positive volume, it follows from 8 that if and only if the vectors u, v, and

w lie in the same plane. Thus we have the following result.

THEOREM 3.4.5

If the vectors , , and have the same initial point, then they lie in the same
plane if and only if

Independence of Cross Product and Coordinates

Initially, we defined a vector to be a directed line segment or arrow in 2-space or 3-space; coordinate systems and components
were introduced later in order to simplify computations with vectors. Thus, a vector has a “mathematical existence” regardless of