Quaternions

Refer to pg. 37

Define a quaternion to be 4-component vector with additional operations (extension of vectors). Has a 3-space vector and an additional scalar component.

$$ \mathbf{q}=\begin{bmatrix}\mathbf{q}{1:3} \\ q_4 \end{bmatrix} \quad \text{where} \quad\mathbf{q}{1:3} \begin{bmatrix} q_1 \\ q_2 \\ q_3 \end{bmatrix} $$

Multiplication

Important quaternion operations are products of quaternions $\bar{\mathbf{q}}$ and $\mathbf{q}$:

$$ \bar{\mathbf{q}} \otimes \mathbf{q} = \begin{bmatrix} q_4 \bar{\mathbf{q}}{1:3} + \bar{q_4}\mathbf{q}{1:3} - \bar{\mathbf{q}}{1:3} \times \mathbf{q}{1:3} \\ \bar{q_4} q_4 - \bar{\mathbf{q}}{1:3} \cdot \mathbf{q}{1:3} \end{bmatrix} $$

$$ \bar{\mathbf{q}} \odot \mathbf{q} = \begin{bmatrix} q_4 \bar{\mathbf{q}}{1:3} + \bar{q_4}\mathbf{q}{1:3} + \bar{\mathbf{q}}{1:3} \times \mathbf{q}{1:3} \\ \bar{q_4} q_4 - \bar{\mathbf{q}}{1:3} \cdot \mathbf{q}{1:3} \end{bmatrix} $$

As these only differ in sign of cross product, the following is true: $\bar{\mathbf{q}}\otimes\mathbf{q}=\mathbf{q}\odot\bar{\mathbf{q}}$

• Multiplication is associative:

$$ \bar{\mathbf{q}} \otimes (\bar{\mathbf{q}} \otimes\bar{\bar{\mathbf{q}}}) = (\mathbf{q} \otimes \bar{\mathbf{q}}) \otimes \bar{\bar{\mathbf{q}}} $$

• Multiplication is distributive:

$$ \mathbf{q} \otimes (\bar{\mathbf{q}} + \bar{\bar{\mathbf{q}}}) = \mathbf{q} \otimes \bar{\mathbf{q}} + \mathbf{q} \otimes \bar{\bar{\mathbf{q}}} $$

• Multiplication is NOT commutative (general case):

$$ \mathbf{q} \otimes \bar{\mathbf{q}} \neq \bar{\mathbf{q}} \otimes \mathbf{q} $$

• If $\mathbf{q}$ and $\bar{\mathbf{q}}$ commute (special case), then $\mathbf{q} \otimes \bar{\mathbf{q}} = \bar{\mathbf{q}} \otimes \mathbf{q}$

<aside> 💡 There are analogous equations for the $\odot$ product

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