I tried a little net search, and came up with

http://mathworld.wolfram.com/RotationFormula.html

I putn=Uandr=(x,y,z), and after some brief computations, formula (3) of that page is exactlyr'=Ar, whereAis the matrix given.

Tell me if the derivation of the formula seems difficult (...not that I will be able to help :P)