Abstract
In our paper we find all functions \({f : \mathbb {R} \times \mathbb {R}^{3} \rightarrow \mathbb {H}}\) and \({g : \mathbb {R}^{3} \rightarrow \mathbb {H}}\) satisfying \({f (r, {\bf v}) f (s, {\bf w}) = -\langle{\bf v},{\bf w}\rangle + f (rs, s{\bf v} + r{\bf w} + {\bf v} \times {\bf w})}\) \({(r, s \in \mathbb {R}, {\bf v},{\bf w} \in \mathbb {R}^{3})}\) , and \({g({\bf v})g({\bf w}) = -\langle{\bf v}, {\bf w}\rangle + g({\bf v} \times {\bf w})}\) \(({{\bf v},{\bf w} \in \mathbb {R}^{3}})\) . These functional equations were motivated by the well-known identities for vector products and quaternions, which can be obtained from the solutions f (r, (v 1, v 2, v 3)) = r + v 1 i + v 2 j + v 3 k and g((v 1 ,v 2, v 3)) = v 1 i + v 2 j + v 3 k.
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Research was supported in part by Grant 75566 from the Hungarian Scientific Research Fund.
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Nyul, B., Nyul, G. Functional equations for vector products and quaternions. Aequat. Math. 85, 35–39 (2013). https://doi.org/10.1007/s00010-012-0120-7
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DOI: https://doi.org/10.1007/s00010-012-0120-7