Find a vector in cartesian coordinates given its relative location to another vector in spherical coordinates

Question

Here is my problem:

-I have an arbitrary normalized vector N in cartesian coordinates

-I am trying to find normalized vector M, also in cartesian coordinates

-I am given the azimuth and polar angles theta and phi of vector M relative to N

How do I do this conversion? I know how to convert between cartesian and spherical coordinates when it is based around the axes but I don't know how to do it about an arbitrary vector N.

Answer 1

If you know how to convert from polar to cartesian coordinates, you only miss to observe one thing. What you called the azimuth and polar angles theta and phi of vector M relative to N are the polar coordinates of (it seems to me to understand) $N+M$. In this case, you convert it to cartesian, and then subtract $N$ componentwise.