Intersection of Two Planes
Two arbitrary planes may be parallel, intersect or coincide:
|
|
|
Parallel Planes |
Intersecting Planes |
Coincident Planes |
-
Parallel planes: Parallel planes are planes that never cross. The ceiling of a room (assuming it’s flat) and the floor are parallel planes (though true planes extend forever in all directions).
-
Intersecting planes: Intersecting planes are planes that cross, or intersect. When planes intersect, the place where they cross forms a line. The floor and a wall of a room are intersecting planes, and where the floor meets the wall is the line of intersection of the two planes.
- Coincident planes: Two planes are coincident when they are the same plane. In general, two planes are coincident if the equation of one can be rearranged to be a multiple of the equation of the other
How to find the relationship between two planes.
Given two 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 the two planes can be described as follow:
| Position |
r |
r' |
|
| Intersecting |
2 |
2 |
 |
| Parallel |
1 |
2 |
 |
| Coincident |
1 |
1 |
 |
State the relationship between the planes:
Therefore r=2 and r'=2. They are Intersecting Planes.
State the relationship between the planes:
Parallel Planes
State the relationship between the planes:
Coincident Planes
