Page 344 - Matrix Analysis & Applied Linear Algebra

340 Chapter 5 Norms, Inner Products, and Orthogonality

and u × v is a vector orthogonal to both u and v. The direction of
u × v is determined from the so-called right-hand rule as illustrated in
Figure 5.6.8.

Figure 5.6.8
Assume the origin is interior to the polytope, and consider a particular
face and three vertices p 0 , p 1 , and p 2 on the face that are positioned
as shown in Figure 5.6.9. The vector n =(p 1 − p 0 ) × (p 2 − p 1 )is
orthogonal to the face, and it points in the outward direction.

Figure 5.6.9
Explain why the outside of the face is visible from the perspective indi-
cated in Figure 5.6.6 if and only if the first component of the outward
normal vector n is positive. In other words, the face is drawn if and
only if n 1 > 0.