Intersection of Two Planes

Given two planes:

Plane Equation

Plane Equation

Form a system with the equations of the planes and calculate the ranks.

System of Equations

r = rank of the coefficient matrix.

r'= rank of the augmented matrix.

The relationship between the two planes can be described as follows:

Position r r'  
Intersecting 2 2 Intersecting Planes
Parallel 1 2 Parallel Planes
Coincident 1 1 Coincident Planes

Relationship between Two Planes


Examples

State the relationship between the planes.

1. Intersection of Two Planes Example

Matrix

Coefficient Matrix

For intersecting planes, the intersection is a line.

System of Equations

Solving the System

Equation of a Line


2. Intersection of Two Planes Example

Parallel Planes

Parallel planes.


3. Intersection of Two Planes Example

Coincident Planes

Coincident planes.