Rotation (mathematics)

Euler rotations of the Earth. Intrinsic (green), precession (blue) and nutation (red)

A three-dimensional rotation can be specified in a number of ways. The most usual methods are:

and equating real and imaginary parts gives the same result as a two-dimensional matrix:

This is multiplied by a vector representing the point to give the result

Above-mentioned Euler angles and axis–angle representations can be easily converted to a rotation matrix.

Another possibility to represent a rotation of three-dimensional Euclidean vectors are quaternions described below.