Quaternions

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Written as

$$q=\left[\begin{array}{c}\eta \\ ϵ\end{array}\right]$$

Where $\eta \in \mathbb{R}$ is called the scalar part and $ϵ\in {\mathbb{R}}^3$ is called the vector part. Quaternion addition is the same as normal vectors, but multiplication is given by

$$q_a\otimes q_b=\left[\begin{array}{c}{\eta }_a{\eta }_b-{ϵ}_a^T{ϵ}_b\\ {\eta }_b{ϵ}_a+{\eta }_a{ϵ}_b+{ϵ}_a\times {ϵ}_b\end{array}\right]$$

Euler Angles to Quaternions