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Maths Exercices

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:

 

1 1
3 -1

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


Planes = 7x-9y+z-9=0 and = 7x-9y+z+5=0 are:

  1. Parallel
  2. Coincident
  3. Intersecting