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Planes in the Space

# 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: Intersecting at a point Each Plane Cuts the Other Two in a Line Three Planes Intersecting in a Line Three Parallel Planes Two Coincident Planes and the Other Parallel Three Coincident Planes