Foot of Perpendicular from a Point to a Plane

Foot of Perpendicular from a Point to a Plane

For point:

$$P(x_1,y_1,z_1)$$

and plane:

$$Ax+By+Cz+D=0$$

let:

$$S=Ax_1+By_1+Cz_1+D$$

Then the foot of perpendicular is:

$$H\left(x_1-\frac{AS}{A^2+B^2+C^2},y_1-\frac{BS}{A^2+B^2+C^2},z_1-\frac{CS}{A^2+B^2+C^2}\right)$$

Usage

- Used to find the perpendicular projection of a point onto a plane.