June 26, 2019

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.

The way this article explained about the matrix is fabulous.. students who have passion in maths definitely like this article

Perfectly explained the topic simply. nice work

So you you have 8 possible arrangements of 3 planes. How many similar arrangements are there of four 4-dimensional hyperplanes. How many similar arrangements of n n-dimensional hyperplanes?

Oops, I meant four 3-dimensional hyperplanes, and n (n-1)-dimensional hyperplanes.

8 arrangements of 3 planes. How many arrangements of 4 3-dimensional hyperplanes? How many arrangements of n (n-1)-dimensional hyperplanes?