Chapters

- 1. Intersecting at a Point
- 2.1 Each Plane Cuts the Other Two in a Line.
- 2.2 Two Parallel Planes and the Other Cuts Each in a Line
- 3.1 Three Planes Intersecting in a Line
- 3.2 Two Coincident Planes and the Other Intersecting Them in a Line
- 4.1 Three Parallel Planes
- 4.2 Two Coincident Planes and the Other Parallel
- 5. Three Coincident 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

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.

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