Perpendicular distance from a 3D point to a vector in spherical polar coordinates.

Question

I have a point $(r, \theta, \phi)$ and a direction vector with angles $(\theta', \phi')$. What would be the method to calculate the shortest distance from the point to the vector?

Answer 1

The Euclidean distance is $$ d = \sqrt{(r \sin \theta \cos \phi - \sin \theta' \cos \phi')^2 + (r \sin \theta \sin \phi - \sin \theta' \sin \phi')^2 + (r \cos \theta - \cos \theta')^2} $$ As the point and vector are given, there is only this distance, which is the shortest by lack of alternatives.