Unit quaternions

A unit quaternion representing rotation can be expressed as:

$$q=\left[\begin{array}{c}{q}_w\\ {\mathbf{q}}_v\end{array}\right]=\left[\begin{array}{c}\cos (\theta /2)\\ \mathbf{u}\sin (\theta /2)\end{array}\right]$$

Where:

• $q_w$ is the scalar part
• ${\mathbf{q}}_v=[q_x,q_y,q_z{]}^T$ is the vector part
• $\theta$ is the rotation angle
• $\mathbf{u}$ is the unit vector representing the rotation axis