Intersection of Three Planes
In 3D, three planes
,
and
can intersect (or not) in the following ways:



All three planes are parallel 

Just two planes are parallel, and
the 3rd plane cuts each in a line




The intersection of the three planes is a line


The intersection of the three planes is a point




Each plane cuts the other two in a line 

Two Coincident Planes and the Other Intersecting Them in a Line 
How to find the relationship between two planes.
Given three planes:
Form a system with the equations of the planes and calculate the ranks.
r = rank of the coefficient matrix
r'= rank of the augmented matrix
The relationship between the two planes can be described as follow:
Case 1. Intersecting at a point
r=3 and r'=3






Case 2.1. Each Plane Cuts the Other Two in a Line
r=2 and r'=3
The three planes form a prismatic surface


Case 2.2. Two Parallel Planes and The Other Cuts Each in a Line
r=2 and r'=3
Two rows of the coefficient matrix are proportional:




Case 3.1. Three Planes Intersecting in a Line
r=2 and r'=2


Case 3.2. Two Coincident Planes and the Other Intersecting Them in a Line
r=2 and r'=2
Two rows of the augmented matrix are proportional:




Case 4.1. Three Parallel Planes
r=1 and r'=2


Case 4.2. Two Coincident Planes and the Other Parallel
r=1 and r'=2
Two rows of the augmented matrix are proportional:




Case 5. Three Coincident Planes
r=1 and r'=1



State the relationship between the three planes:
Solution:
Form a system with the equations of the planes and calculate the ranks.
Each plane cuts the other two in a line and they form a prismatic surface.
State the relationship between the three planes:
Solution:
Form a system with the equations of the planes and calculate the ranks.
Each plane intersects at a point.
State the relationship between the three planes:
Solution:
Form a system with the equations of the planes and calculate the ranks.
The Second and Third planes are Coincident and the first is cutting them, therefore the three planes intersect in a line.