Shortest Distance from a Point to a Line

• The shortest distance between two points is always the line that makes a $90°$ angle with both the surfaces.
• Since perp is perpendicular to the projected upon surface, we can use this.

$$\begin{array}{r}\overrightarrow{RQ}={\text{perp}}_{\vec{d}}(\overrightarrow{PR})\end{array}$$

Point $Q$ can be found by:

$$\begin{array}{r}\overrightarrow{OQ}=\overrightarrow{OP}+{\text{proj}}_{\vec{d}}(\overrightarrow{PR})\end{array}$$