Distances in the space

Plane-Plane Distance

If you have infinite planes, the only way they don't intersect is if they are parallel which makes the distance between them easy.

DISTANCE BETWEEN PARALLEL PLANES
The distance between two parallel planes is the minimum distance from any point from one plane to a point on the other plane.

To determine the distance between two parallel planes

$\pi_1\;\equiv\;Ax+By+Cz+D_1=0\;$ and $\pi_2\;\equiv\;Ax+By+Cz+D_2=0\;$

we can use the formula:

$d(\pi_1,\pi_2)=\;\frac{|D_2-D_1|}{sqrt{A^2+B^2+C^2}}\;$

Calculate the distance between the planes:

$\pi_1\;\equiv\;2x-y-2z+5=0\;$ and $\pi_2\;\equiv\;4x-2y-4z+15=0\;$

Solution:

$\frac{2}{4}=\frac{-1}{-2}=\frac{-2}{-4}\neq\frac{5}{15}\;$

The two planes are parallel.

Transform the equation of the second plane so that the two planes have the same normal vector:

$\pi_2\;\equiv\;x-y-2z+\frac{15}{2}=0\;$

$d(\pi_1,\pi_2)=\;\frac{|\frac{15}{2}-5|}{sqrt{2^2+(-1)^2+(-2)^2}}=\frac{5}{6}\;$