Foot of Perpendicular from a Point to a Line

Foot of Perpendicular from a Point to a Line

For:

$$\vec r=\vec a+\lambda\vec d$$

and point $$p$$ with position vector $$\vec p$$, let the foot be $$h$$:

$$\vec h=\vec a+\lambda\vec d$$

The perpendicularity condition is:

$$(\vec p-\vec h)\cdot\vec d=0$$

Hence:

$$\lambda=\frac{(\vec p-\vec a)\cdot\vec d}{|\vec d|^2}$$

Usage

- Used to locate the perpendicular projection of a point onto a line.