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, $\text{d}(\pi, \pi') = 0$.
• 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. $$\text{d}(\pi,\pi') = \text{d}(P,\pi') = \text{d}(\pi,P')$$ where $P\in\pi$ and $P'\in\pi'$.

Find the distance between the following planes:

$$\pi: 2x - 4y + 4z +3 = 0 \qquad \pi': x - 2y + 2z -1 = 0$$

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

Yes, they are.

Therefore, we can take the point $P'= (1, 0, 0)$ belonging to $\pi'$ and do: $$\text{d}(\pi,\pi')=\text{d}(P',\pi) = \dfrac{|2\cdot1-4\cdot0+4\cdot0+3|}{\sqrt{2^2+(-4)^2+4^2}}= \dfrac{5}{6}$$

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

$$\pi: Ax + By + Cz + D = 0 \qquad \pi': Ax + By + Cz + D' = 0$$

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

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