Distance from a point to a plane in space

The distance between a point $P$ and a plane $π$ , $d (P, π)$ , is the minimal distance between $P$ and any point of the plane.

If $P$ is a point of the plane $π$ , the distance is zero.
If $P$ is not a point of the plane $π$ , the distance is the module of $\vec{P P^{'}}$ , where $P^{'}$ the orthogonal projection of $P$ on the plane $π$ .

Nevertheless, there exists a much more practical formula (but not easy to obtain) that is presented next:

Let $P = (p_{1}, p_{2}, p_{3})$ and let $π : A x + B y + C z + D = 0$ . Then,

$d (P, π) = \frac{| A \cdot p_{1} + B \cdot p_{2} + C \cdot p_{3} + D |}{\sqrt{A^{2} + B^{2} + C^{2}}}$

Example

Calculate the distance between the point $P = (- 2, 0, 3)$ and the plane $π : 4 x + 2 y - 4 z + 3 = 0$ .

We can apply the formula: $d (P, π) = \frac{| A \cdot p_{1} + B \cdot p_{2} + C \cdot p_{3} + D |}{\sqrt{A^{2} + B^{2} + C^{2}}} = \frac{| 4 \cdot (- 2) + 2 \cdot 0 - 4 \cdot 3 + 3 |}{\sqrt{4^{2} + 2^{2} + (- 4)^{2}}} = \frac{17}{6}$