This is a continuation of a former work co-authored by the author on the connection between product of quaternions and product of three-dimensional vectors. In this paper, all the solutions of the functional equations \[ g_{1}(x)g_{2}(y)=-\langle x,y\rangle+g_{3}(x\times y) \qquad (x,y\in\mathbb{R}^{3}) \] and \[ f_{1}(r,x)f_{2}(s,y)=-\langle x,y\rangle+f_{3}(rs,sx+ry+x\times y) \qquad (r,s\in\mathbb{R}, x,y\in\mathbb{R}^{3}) \] are determined. Here \(g_{1},g_{2},g_{3}:\mathbb{R}^{3}\to \mathbb{H}\) and \(f_{1},f_{2},f_{3}:\mathbb{R}\times\mathbb{R}^{3}\to \mathbb{H}\) the unknown functions while \(\mathbb{R}\) and \(\mathbb{H}\) denote the set of real numbers and the set of the skew field of quaternions, respectively. Special cases and consequences are also investigated.