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, (v1, v2, v3)) = r + v1i + v2j + v3k and g((v1 ,v2, v3)) = v1i + v2j + v3k.