# 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.