Distance between two planes in space

To calculate the distance between any two planes, it is necessary to bear in mind its relative position:

If the planes coincide or are secant, the distance between them is zero, $d (π, π^{'}) = 0$ .
If the planes are parallel, the distance between them is calculated taking any point of one plane and calculating the distance between this choosen point and the other plane. $d (π, π^{'}) = d (P, π^{'}) = d (π, P^{'})$ where $P \in π$ and $P^{'} \in π^{'}$ .

Example

Find the distance between the following planes:

$π : 2 x - 4 y + 4 z + 3 = 0 π^{'} : x - 2 y + 2 z - 1 = 0$

We verify that the planes are parallel: $\frac{2}{1} = \frac{- 4}{- 2} = \frac{4}{2}$

Yes, they are.

Therefore, we can take the point $P^{'} = (1, 0, 0)$ belonging to $π^{'}$ and do: $d (π, π^{'}) = d (P^{'}, π) = \frac{| 2 \cdot 1 - 4 \cdot 0 + 4 \cdot 0 + 3 |}{\sqrt{2^{2} + (- 4)^{2} + 4^{2}}} = \frac{5}{6}$

Another good way of calculating the distance between parallel planes. If we have them expressed as follows:

$π : A x + B y + C z + D = 0 π^{'} : A x + B y + C z + D^{'} = 0$

It consists in using its distance to the origin of coordinates, which allows us to obtain the following expression:

$d (π, π^{'}) = \frac{| D - D |}{\sqrt{A^{2} + B^{2} + C^{2}}}$