I'm trying to create a random uniformly distributed vector through a
hemisphere positioned at the intersection point between ray and an
object. I can already create a random vector but how do I quickly/easily
rotate it based on the desired hemisphere orientation?

I use left-handed coordinatesystem: x is right, y is up, and z is forward.

Let's have a vector V which is random vector inside a hemisphere which
has a pole at 0,1,0 (just like the northern hemisphere of Earth). I want
to orient the vector inside a hemishere which has a pole pointing at
vector N. How do I do this? I'd prefer something that can be done
without expensive trigonometric functions - if possible.

PS. If this is the wrong forum, please advice.

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