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# Rotating a Unit Vector in 3D Using Quaternions

A quaternion is a vector in with a noncommutative product (see [1] or Quaternion). Quaternions, also called hypercomplex numbers, were invented by William Rowan Hamilton in 1843. A quaternion can be written or, more compactly, or , where the noncommuting unit quaternions obey the relations .
A quaternion can represent a rotation axis, as well as a rotation about that axis. Instead of turning an object through a series of successive rotations using rotation matrices, quaternions can directly rotate an object around an arbitrary axis (here ) and at any angle . This Demonstration uses the quaternion rotation formula with , a pure quaternion (with real part zero), , normalized axis , and for a unit quaternion, , where the quaternion conjugate for is .

### DETAILS

Reference
[1] Wikipedia. "Quaternions and Spatial Rotation." (Feb 9, 2016) en.wikipedia.org/wiki/Quaternions_and_spatial _rotation.

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