Distance from a point to a plane

How far a point sits from a plane, found by projecting onto the plane's unit normal.

The distance from a point $P_1$ to a plane is found by building any vector from a known plane point to $P_1$ , then projecting that vector onto the unit normal $\hat{\mathbf{n}}$ — a scalar projection in disguise. The absolute value strips away which side of the plane $P_1$ sits on, since distance is never negative.

Examples

The distance from $P_1=(2,-3,5)$ to the plane $2x-y+3z=6$ is $D=\dfrac{|22-6|}{\sqrt{14}}=\dfrac{16}{\sqrt{14}}\approx 4.28$ .
A robot arm's path planner checks the distance from every joint to nearby flat surfaces with exactly this formula, thousands of times per second.