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I came accross different notations for the multiplication between two quaternions, e.g:

\begin{equation} \mathbf{q}_1 \circ \mathbf{q}_2 \quad \text{or} \quad \mathbf{q}_1 \otimes \mathbf{q}_2 \end{equation}

Which one should be preferred? Is there any standard notation for this operation ? The latter seems to be widely used for the outer product.

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    $\begingroup$ The other option is no symbol between the quaternions. $\endgroup$
    – J.G.
    Sep 30, 2022 at 11:20
  • $\begingroup$ @J.G. well if this notation is considered as the standard notation for quaternion multiplication I'm ready to accept the answer $\endgroup$
    – Gab
    Sep 30, 2022 at 11:34
  • $\begingroup$ @ParclyTaxel for the first notation here for example, for the second one I saw it on multiple websites when searching for an answer prior to post this question $\endgroup$
    – Gab
    Sep 30, 2022 at 11:35
  • $\begingroup$ Semanticscholar, bioRxiv, Montana State University. $\endgroup$
    – Gab
    Sep 30, 2022 at 11:38
  • $\begingroup$ here and here, I cannot find the one on semanticscholar $\endgroup$
    – Gab
    Sep 30, 2022 at 11:42

1 Answer 1

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$\mathbf q_1\circ\mathbf q_2$ recalls the interpretation of certain quaternions as rotations in 3D space, the composition of functions mapping said space to itself. $\mathbf q_1\otimes\mathbf q_2$ is a reminder that multiplication of quaternions is noncommutative.

But the most common and most concise notation is simply $\mathbf q_1\mathbf q_2$ – the quaternions being a skew-field where multiplication is defined, albeit a noncommutative one, is more than enough to merit using juxtaposition to denote multiplication. (This is also very common in group theory.)

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  • $\begingroup$ Thank you for your answer, I appreciate the details on all notations $\endgroup$
    – Gab
    Sep 30, 2022 at 11:54

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