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Maths Exercices

Intersection of Three Planes

In 3D, three planes , and can intersect (or not) in the following ways:

 
All three planes are parallel
 
Just two planes are parallel, and the 3rd plane cuts each in a line
 

The intersection of the three planes is a line

 

 

The intersection of the three planes is a point

 



 
Each plane cuts the other two in a line   Two Coincident Planes and the Other Intersecting Them in a Line

How to find the relationship between two planes.

Given 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 the two planes can be described as follow:

Case 1. Intersecting at a point
r=3 and r'=3

   
     



Case 2.1. Each Plane Cuts the Other Two in a Line
r=2 and r'=3

The three planes form a prismatic surface

 

 

 

Case 2.2. Two Parallel Planes and The Other Cuts Each in a Line
r=2 and r'=3
Two rows of the coefficient matrix are proportional:

     

Case 3.1. Three Planes Intersecting in a Line
r=2 and r'=2




 

Case 3.2. Two Coincident Planes and the Other Intersecting Them in a Line
r=2 and r'=2
Two rows of the augmented matrix are proportional:

     

Case 4.1. Three Parallel Planes
r=1 and r'=2

 

Case 4.2. Two Coincident Planes and the Other Parallel
r=1 and r'=2
Two rows of the augmented matrix are proportional:

     

Case 5. Three Coincident Planes
r=1 and r'=1

   

 



State the relationship between the three planes:


Solution:
Form a system with the equations of the planes and calculate the ranks.

1 1 -1
-4 1 4
-2 3 2
=0 r=2

 

1 1 -3
-4 1 7
-2 3 2
r'=3

Each plane cuts the other two in a line and they form a prismatic surface.


State the relationship between the three planes:


Solution:
Form a system with the equations of the planes and calculate the ranks.

2 3 4
1 -1 -1
-1 2 -1
r=3

 

2 3 4
1 -1 -1
-1 2 -1
r'=3

Each plane intersects at a point.


State the relationship between the three planes:


Solution:
Form a system with the equations of the planes and calculate the ranks.

2 3 1
1 -1 1
2 -2 2
=0 r=2

 

2 3 1
1 -1 -2
2 -2 -4
=0 r'=2

The Second and Third planes are Coincident and the first is cutting them, therefore the three planes intersect in a line.


The planes : -6x+5y+2z=-7 , : -5x+3y+4z=3 and : -x+2z=-8 are:

  1. Intersecting at a point
  2. Each Plane Cuts the Other Two in a Line
  3. Three Planes Intersecting in a Line
  4. Three Parallel Planes
  5. Two Coincident Planes and the Other Parallel
  6. Three Coincident Planes