Intersection of Three Planes

To study the intersection of three planes, form a system with the equations of the planes and calculate the ranks.

Three Planes

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

Planes Intersecting at a Point

2.1 Each Plane Cuts the Other Two in a Line.

r = 2, r' = 3

The three planes form a prismatic surface.

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.

Two Parallel Planes and an Intersecting Plane

Two Parallel Planes and an Intersecting Plane

3.1 Three Planes Intersecting in a Line

r = 2, r' = 2

Three Planes Intersecting in a Line

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.

Two Coincident Planes and an Intersecting Plane

Two Coincident Planes and an Intersecting Plane

4.1 Three Parallel Planes

r = 1, r' = 2

Three Parallel Planes

4.2 Two Coincident Planes and the Other Parallel

r = 1, r' = 2

Two rows of the augmented matrix are proportional.

Two Coincident Planes and a Parallel Plane

Two Coincident Planes and a Parallel Plane

5. Three Coincident Planes

r = 1, r' = 1

Three Coincident Planes


Examples

State the relationship between the three planes.

1. Relationship between Three Planes Example

System of Equations

Coefficient Matrix

Augmented Matrix

Each plane cuts the other two in a line and they form a prismatic surface.

2. Relationship between Three Planes Example

System of Equations

Coefficient Matrix

Augmented Matrix

Each plan intersects at a point.

3. Relationship between Three Planes Example

System of Equations

Coefficient Matrix

Augmented Matrix

Proportion

The second and third planes are coincident and the first is cuting them, therefore the three planes intersect in a line.

4. Relationship between Three Planes Example

System of Equations

Coefficient Matrix

Augmented Matrix

Proportion

The first and second are coincident and the third is parallel to them.