'the path of the tip of rotated vectors' or 'paths on a sphere'

I need to plot the path the tip of a specific vector draws,
when the vector is rotated around certain axis for some angels

as a basic example:
start vector is (0,0,1)
rotation around axis (0,1,0) for an angle of (pi/2)
rotation around axis (0,0,1) for an angle of (pi)

so the path should show something like starting at the bottom,
go up to the equator plane and then turn around the equator
to the other side

this could also be expressed in terms of a point moving on
a sphere ...

a graphical example would be like the path C in this figure

does anyone happen to know how i could easily implement something
like that?

with best regards
and thanks for your help/suggestions


The king who needs to remind his people of his rank, is no king.

A beggar's mistake harms no one but the beggar. A king's mistake,
however, harms everyone but the king. Too often, the measure of
power lies not in the number who obey your will, but in the number
who suffer your stupidity.