:PROPERTIES: :ID: 2c9478e0-4c97-45d0-b4e6-9003425e2c3c :mtime: 20231017010409 20231014011418 :ctime: 20231014011329 :END: #+title: shortest distance between 2 lines #+filetags: :public:project: * Definition Given 2 lines, return the shortest distance between them * Vector Method Let $\vec{r} = \vec{a} + \lambda \vec{t}$ and let $\vec{s} = \vec{b} + \mu \vec{u}$. The line joining them will be parallel to $\vec{t} \times \vec{u}$. Using [[id:856a2afe-f299-4eb8-b0ef-018aa5256d8c][shortest distance from line to a point]], we get the distance as \[d = \frac{|(\vec{b} - \vec{a}) \cdot (\vec{t} \times \vec{u})|}{|\vec{t} \times \vec{u}|} \] Note that the pattern of $\cdot$ followed by $\times$ is called [[id:e72ab120-e332-4c9d-bf88-ea02dcd4eb99][scalar triple product.]] ** When do 2 lines intersect? 2 lines interesect iff $d = 0$, wich means they intersect iff \[(\vec{b} - \vec{a}) \cdot (\vec{t} \times \vec{u}) = 0\] ** Shortest distance between 2 parallel lines if 2 lines are parallel, the above method results in dividing by 0, so \[d = \frac{|(\vec{b} - \vec{a}) \times \vec{t}|}{|\vec{t}|}\]