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So let's say I have a normalized vector $N$ given in cartesian coordinates and I have another normalized vector $V$, defined in spherical coordinates relative to the vector $N$. So $\theta_V$ is the angle of $V$ with $N$ and $\phi_V$ is the rotation around $N$.

How would I convert these spherical coordinates to cartesian coordinates? I know how the transformation works normally but I don't know how it works when you're given a specific vector as a reference.

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You can reduce it to a problem you know how to solve by rotating the system so that $N$ is at the North Pole, doing the transformation, and then rotating back.

Note that however you approach this, you currently have a degree of freedom because you haven't defined the principal meridian.

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