Quaternions and spatial rotation

Summary

Unit quaternions describe rotations of three-dimensional space. A unit quaternion $q$ acts on a vector $v$, viewed as a pure quaternion, by $v \mapsto q v \bar q$, and this map is a rotation. The quaternion $\cos(\theta/2) + \sin(\theta/2)(u_1 i + u_2 j + u_3 k)$ gives the rotation by $\theta$ about the unit axis $u$, and $q$ and $-q$ give the same rotation. This description avoids the gimbal lock of Euler angles and is used in mechanics, robotics, computer graphics and spacecraft attitude control.