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Planes in the Space
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:

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

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

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

$\left{A_1x+B_1y+C_1z+D_1=0\\A_2x+B_2y+C_2z+D_2=0$

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 $\frac{A_1}{A_2}\;\neq\;\frac{B_1}{B_2}\;\neq\;\frac{C_1}{C_2}$ Parallel 1 2 $\frac{A_1}{A_2}\;=\;\frac{B_1}{B_2}\;=\;\frac{C_1}{C_2}\;\neq\;\frac{D_1}{D_2}$ Coincident 1 1 $\frac{A_1}{A_2}\;=\;\frac{B_1}{B_2}\;=\;\frac{C_1}{C_2}\;=\;\frac{D_1}{D_2}$

State the relationship between the planes:

$\left{x+y-5z+4=0\\3x-y+15z-1=0$

 $M_1=\left(\begin{matrix}1&1&-5\\3&-1&15\\\end{matrix}\right)$ $M_2=\left(\begin{matrix}1&1&-5&-4\\3&-1&15&1\\\end{matrix}\right)$

 1 1 3 -1
$\neq\;0$

Therefore r=2 and r'=2. They are Intersecting Planes.

State the relationship between the planes:

$\left{x+y-5z+4=0\\-3x-3y+15z-1=0$

$\frac{1}{-3}\;=\;\frac{1}{-3}\;=\;\frac{-5}{15}\;\neq\;\frac{4}{-1}$

Parallel Planes

State the relationship between the planes:

$\left{x+y-5z+4=0\\-3x-3y+15z-12=0$

$\frac{1}{-3}\;=\;\frac{1}{-3}\;=\;\frac{-5}{15}\;\neq\;\frac{4}{-12}$

Coincident Planes

 Planes = -9x-6y-2z-1=0 and = 9x+6y+2z-9=0 are: Parallel Coincident Intersecting