Abstract. In our paper we find all functions f : R × R 3 → H and g : R 3 → H satisfying f (r, v)f (s, w) = − v, w + f (rs, svThese 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)
Mathematics Subject Classification (2010). 39B52, 16K20.Keywords. functional equation, vector product, quaternion.
Quaternions and vector productsLet H = {r+v 1 i+v 2 j +v 3 k | r, v 1 , v 2 , v 3 ∈ R} be the skew field of quaternions with the basic relations i 2 = j 2 = k 2 = ijk = −1. We call r the real part and v 1 i + v 2 j + v 3 k the imaginary part of the quaternion r + v 1 i + v 2 j + v 3 k ∈ H. A quaternion is purely imaginary if its real part equals 0. The absolute value of the above quaternion is defined by r 2 + v 2 1 + v 2 2 + v 2 3 . It is well-known (see e.g. [1], [6], [9], [10]) that identifying the purely imaginary quaternion v 1 i+v 2 j +v 3 k with the vector (v 1 , v 2 , v 3 ) ∈ R 3 we have (r + v)(s + w) = (rs − v, w ) + (sv + rw + v × w) (r, s ∈ R, v, w ∈ R 3 ), and for purely imaginary quaternionswhere on the left-hand sides quaternionic products stand, v, w and v × w denote the standard inner product (dot product) and the cross product of v, w ∈ R 3 , respectively.
