Chapter 12, Problem 13AAE

(a)

To determine

Prove that the distance between the parallel planes $Ax+By+Cz=D_1$ and $Ax+By+Cz=D_2$ is $d=\frac{\left|D_1-D_2\right|}{\left|Ai+Bj+Ck\right|}$

(b)

To determine

Calculate the distance between the planes $2x+3y-z=6$ and $2x+3y-z=12$.

(c)

To determine

Calculate an equation for the plane parallel to the plane $2x-y+2z=-4$ if the point $\left(3,2,-1\right)$ is equidistant from the two planes.

(d)

To determine

Find the equations for the planes that lie parallel to, and 5 units away from, the plane $x-2y+z=3$.