Given two planes, discuss the methods used to determine if the planes are parallel, perpendicular, coincident, or none of these.

Answer 1

Given two planes:
Π1 : Ax+By+Cz+D=0 and
Π2 : ax+by+cz+d=0

The following uses the notation that
<a,b,c> represents a vector with three components.

The normal vectors are:
N1 : <A,B,C>
N2 : <a,b,c>

If Π1 is perpendicular to Π2, then
N1.N2=0 (dot-product).

If the planes are parallel, we can find a numeric factor α such that
<A,B,C>=α<a,b,c>
If in addition, D=αd, then the planes are coincident.

If no such α can be found, the planes are not parallel.