Page 244 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Problems Involving Distance

We conclude this section by discussing two basic “distance problems” in 3-space:

Problems
(a) Find the distance between a point and a plane.
(b) Find the distance between two parallel planes.

The two problems are related. If we can find the distance between a point and a plane, then we can find the distance between
parallel planes by computing the distance between either one of the planes and an arbitrary point in the other (Figure 3.5.6).

Figure 3.5.6
The distance between the parallel planes V and W is equal to the distance between and W.

THEOREM 3.5.2

Distance Between a Point and a Plane

The distance D between a point and the plane is

(9)

Proof Let be any point in the plane. Position the normal so that its initial point is at Q. As illustrated

in Figure 3.5.7, the distance D is equal to the length of the orthogonal projection of on n. Thus, from (10) of Section 3.3,

But