Distance of a Point from a Line (3D)

Vector Form: 

The length of the perpendicular from a point P (α) to the line \[\overline{\mathrm{r}}=\overline{\mathrm{a}}+\lambda\overline{\mathrm{b}}\] is

\[\sqrt{\left|\overline{\alpha}-\overline{a}\right|^2-\left[\frac{(\overline{\alpha}-\overline{a}).\overline{b}}{\left|\overline{b}\right|}\right]^2}\]

Cartesian Form:

The length of the perpendicular from the point P (a, b, c) on the line \[\frac{x-x_{1}}{l}=\frac{y-y_{1}}{\mathrm{m}}=\frac{\mathrm{z-z_{1}}}{\mathrm{n}}\] is \[\sqrt{[(\mathrm{a-x_{1}})^{2}+(\mathrm{b-y_{1}})^{2}+(\mathrm{c-z_{1}})^{2}]-[(\mathrm{a-x_{1}})l+(\mathrm{b-y_{1}})\mathrm{m+(c-z_{1})n}]^{2}}\] where l, m, n are direction cosines of line.