Functional equations for vector products and quaternions. (English) Zbl 1271.39018
Motivated by certain identities in the skew field \(\mathbb{H}\) of quaternions, the authors consider and solve completely the functional equations
\[
f(r,v)\cdot f(s,w)=-\langle v,w\rangle+f(r s,s v+r w+v\times w)
\]
and
\[
g(v)\cdot g(w)=-\langle v,w\rangle+g(v\times w),
\]
where \(f:\mathbb{R}\times\mathbb{R}^3\to\mathbb{H}\), \(g:\mathbb{R}^3\to\mathbb{H}\), and \(r,s\in\mathbb{R}\), \(v,w\in\mathbb{R}^3\).
Reviewer: Jens Schwaiger (Graz)
MSC:
| 39B52 | Functional equations for functions with more general domains and/or ranges |
| 16K20 | Finite-dimensional division rings |
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