- - Conjugated quaternion \(q^{\star}=(-x,-y,-z,w)\)
- - Quaternion norm \(\|q\|=\sqrt{q\,q^{\star}} = \sqrt{x^2+y^2+z^2+w^2}\). Unit quaternion satisfies \(\|q\|=1\).
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- \(q_1\;q_2=(x_1\,{\bf i}+ y_1 \, {\bf j} + z_1 \, {\bf k} + w_1)\;(x_2\,{\bf i}+ y_2 \, {\bf j} + z_2 \, {\bf k} + w_2) = \; ...\)
\( q_1 \; q_2 = \left( \begin{array}{r} x_1 w_2 + w_1 x_2 + y_1 z_2 - z_1 y_2 \\ y_1 w_2 + w_1 y_2 + z_1 x_2 - x_1 z_2 \\
z_1 w_2 + w_1 z_2 + x_1 y_2 - y_1 x_2 \\ w_1 w_2 - x_1 x_2 - y_1 y_2 - z_1 z_2 \end{array} \right) \)
- Sometimes interesting to separate real part \(w\) from pure quaternion part \(s=(x,y,z)\).
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- - Shorthand vector form \(q=(s,w)\)
- - Quaternion product in vector form \(q_1\;q_2=(s_1\,w_2+s_2\,w_1+s_1\times s_2 \;,\;w_1\,w_2-s_1\cdot s_2)\)
Note: Quaternion product is
- - associative: \((q_1 q_2) q_3 = q_1 (q_2 q_3) = q_1\, q_2\, q_3\)
- - non-commutative: \(q_1\, q_2 \neq q_2\, q_1\)