Intersection of Three Planes
To study the intersection of 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 three planes presents can be described as follows:
1. Intersecting at a Point
r=3, r'=3
2.1 Each Plane Cuts the Other Two in a Line.
r = 2, r' = 3
The three planes form a prismatic surface.
2.2 Two Parallel Planes and the Other Cuts Each in a Line
r = 2, r' = 3
Two rows of the coefficient matrix are proportional.
3.1 Three Planes Intersecting in a Line
r = 2, r' = 2
3.2 Two Coincident Planes and the Other Intersecting Them in a Line
r = 2, r' = 2
Two rows of the augmented matrix are proportional.
4.1 Three Parallel Planes
r = 1, r' = 2
4.2 Two Coincident Planes and the Other Parallel
r = 1, r' = 2
Two rows of the augmented matrix are proportional.
5. Three Coincident Planes
r = 1, r' = 1
Examples
State the relationship between the three planes.
1. 
Each plane cuts the other two in a line and they form a prismatic surface.
2. 
Each plan intersects at a point.
3. 
The second and third planes are coincident and the first is cuting them, therefore the three planes intersect in a line.
4. 
The first and second are coincident and the third is parallel to them.
