Page 245 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Thus (10)
Since the point lies in the plane, its coordinates satisfy the equation of the plane; thus

or

Substituting this expression in (10) yields (9).

Figure 3.5.7
Distance from to plane.

Remark Note the similarity between (9) and the formula for the distance between a point and a line in 2-space [13 of Section
3.3].

EXAMPLE 8 Distance Between a Point and a Plane

Find the distance D between the point and the plane .

Solution

To apply (9), we first rewrite the equation of the plane in the form

Then

Given two planes, either they intersect, in which case we can ask for their line of intersection, as in Example 6, or they are
parallel, in which case we can ask for the distance between them. The following example illustrates the latter problem.