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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:

$\pi_1\;\equiv\;A_1x+B_1y+C_1z+D_1=0\;$

$\pi_2\;\equiv\;A_2x+B_2y+C_2z+D_2=0\;$

$\pi_3\;\equiv\;A_3x+B_3y+C_3z+D_3=0\;$

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

$\left{A_1x+B_1y+C_1z+D_1=0\\A_2x+B_2y+C_2z+D_2=0\\A_3x+B_3y+C_3z+D_3=0$

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: $\frac{A}{A*}\;=\;\frac{B}{B*}\;=\;\frac{C}{C*}\;\neq\;\frac{D}{D*}$ 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: $\frac{A}{A*}\;=\;\frac{B}{B*}\;=\;\frac{C}{C*}\;=\;\frac{D}{D*}$ 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: $\frac{A}{A*}\;=\;\frac{B}{B*}\;=\;\frac{C}{C*}\;=\;\frac{D}{D*}$ Case 5. Three Coincident Planes r=1 and r'=1

State the relationship between the three planes:

$\pi_1\;\equiv\;x+y-z+3\;=\;0\;$

$\pi_2\;\equiv\;-4x+y+4z-7\;=\;0\;$

$\pi_3\;\equiv\;-2x+3y+2z-2\;=\;0\;$

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

$\left{x+y-z=-3\\-4x+y+4z=7\\-2x+3y+2z=2$

$M_1=\left(\begin{matrix}1&1&-1\\-4&1&4\\-2&3&2\\nd{matrix}\right)$
 1 1 -1 -4 1 4 -2 3 2
=0 r=2

$M_2=\left(\begin{matrix}1&1&-1&-3\\-4&1&4&7\\-2&3&2&2\\nd{matrix}\right)$
 1 1 -3 -4 1 7 -2 3 2
$\neq\;0$ r'=3

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

State the relationship between the three planes:

$\pi_1\;\equiv\;2x-3y+4z-1=0\;$

$\pi_2\;\equiv\;x-y-z+1\;=\;0\;$

$\pi_3\;\equiv\;-x+2y-z+2\;=\;0\;$

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

$\left{2x-3y+4z=1\\x-y-z=-1\\-x+2y-z=-2$

$M_1=\left(\begin{matrix}2&-3&4\\1&-1&-1\\-1&2&-1\\nd{matrix}\right)$
 2 3 4 1 -1 -1 -1 2 -1
$\neq\;0$ r=3

$M_2=\left(\begin{matrix}2&-3&4&1\\1&-1&-1&-1\\-1&2&-1&-2\\nd{matrix}\right)$
 2 3 4 1 -1 -1 -1 2 -1
$\neq\;0$ r'=3

Each plane intersects at a point.

State the relationship between the three planes:

$\pi_1\;\equiv\;2x+3y+z-1=0\;$

$\pi_2\;\equiv\;x-y+z+2\;=\;0\;$

$\pi_3\;\equiv\;2x-2y+2z+4\;=\;0\;$

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

$\left{2x+3y+z=1\\x-y+z=-2\\2x-2y+2z=-4$

$M_1=\left(\begin{matrix}2&3&1\\1&-1&1\\2&-2&2\\nd{matrix}\right)$
 2 3 1 1 -1 1 2 -2 2
=0 r=2

$M_2=\left(\begin{matrix}2&3&1&1\\1&-1&1&-2\\2&-2&2&-4\\nd{matrix}\right)$
 2 3 1 1 -1 -2 2 -2 -4
=0 r'=2

$\frac{1}{2}\;=\;\frac{-1}{-2}\;=\;\frac{1}{2}\;=\;\frac{-2}{-4}$

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

 The planes : -7x+y+5z=9 , : -28x+4y+20z=-36 and : -35x+5y+25z=45 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