Distance from a point to a plane in space

The distance between a point $P$ and a plane $\pi$, $\text{d}(P,\pi)$, is the minimal distance between $P$ and any point of the plane.

• If $P$ is a point of the plane $\pi$, the distance is zero.
• If $P$ is not a point of the plane $\pi$, the distance is the module of $\overrightarrow{PP'}$, where $P'$ the orthogonal projection of $P$ on the plane $\pi$.

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 $\pi: Ax+By+Cz+D = 0$. Then,

$$\text{d}(P,\pi)=\dfrac{|A\cdot p_1+B\cdot p_2+C\cdot p_3+D|} {\sqrt{A^2+B^2+C^2}}$$

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

We can apply the formula: $$\text{d}(P,\pi)=\dfrac{|A\cdot p_1+B\cdot p_2+C\cdot p_3+D|} {\sqrt{A^2+B^2+C^2}} = \dfrac{|4\cdot(-2)+2\cdot0-4\cdot3+3|}{\sqrt{4^2+2^2+(-4)^2}} = \dfrac{17}{6}$$