Distance from a point to a line

How far a point sits from a line in space, found from the cross product: $D=|\overrightarrow{P_0P_1}\times\mathbf{v}|/|\mathbf{v}|$ .

The distance from a point $P_1$ to a line through $P_0$ with direction $\mathbf{v}$ is $D=\dfrac{|\overrightarrow{P_0P_1}\times\mathbf{v}|}{|\mathbf{v}|}$ . The parallelogram spanned by $\overrightarrow{P_0P_1}$ and $\mathbf{v}$ has area $|\overrightarrow{P_0P_1}\times\mathbf{v}|$ , and dividing that area by its base $|\mathbf{v}|$ leaves exactly the parallelogram's height, which is the point's perpendicular distance to the line.

Examples

Swinging a taut string from a point to the nearest spot on a line, snapped perpendicular to it, gives the shortest possible length — exactly what the cross-product formula computes.