Editing Euler's rotation theorem (section)

===Quaternions===
{{Main|Three-dimensional rotation operator|Quaternions and spatial rotation}}

It follows from Euler's theorem that the relative orientation of any pair of coordinate systems may be specified by a set of three independent numbers. Sometimes a redundant fourth number is added to simplify operations with quaternion algebra. Three of these numbers are the direction cosines that orient the eigenvector. The fourth is the angle about the eigenvector that separates the two sets of coordinates. Such a set of four numbers is called a '''[[quaternion]]'''.

While the quaternion described above does not involve [[complex number]]s, if quaternions are used to describe two successive rotations, they must be combined using the non-commutative [[quaternion]] algebra derived by [[William Rowan Hamilton]] through the use of imaginary numbers.

Rotation calculation via quaternions has come to replace the use of [[direction cosines]] in aerospace applications through their reduction of the required calculations, and their ability to minimize [[round-off error]]s. Also, in [[computer graphics]] the ability to perform spherical interpolation between quaternions with relative ease is of value.