Quaternions are used in computer graphics...there are quite a number of questions on this, some I have answered, but got tired of typing the same thing over and over. The short version is, you write a 3-D point as a pure quaternion $v,$ (zero real part). Then, given a unit quaternion $\xi,$ a rotated pure quaternion comes from $\xi v \bar{\xi}.$ I'm not sure these give much help with contest problems. –  Will Jagy Jul 28 '12 at 21:36