:PROPERTIES: :ID: 856a2afe-f299-4eb8-b0ef-018aa5256d8c :mtime: 20231014010653 :ctime: 20231014010556 :END: #+title: shortest distance from line to a point #+filetags: :public:project: * Definiton Given a line and a point, return the shortest distance from the line to the point. * Vector Method Given a line with equation $\vec{r} = \vec{a} + \lambda(\vec{b} - \vec{a})$ and given a point with position $\vec{q}$. the distance is given by $d = \frac{|(\vec{q} - \vec{a}) \times (\vec{b} - \vec{a})|}{|\vec{b} - \vec{a}}$ You could also write this equation as \[d = \frac{|\vec{AQ} \times \vec{AB}}{|\vec{AB}|}\]