Pexiderized functional equations for vector products and quaternions

The purpose of the present paper is to solve the pexiderized versions of functional equations investigated by B. Nyul and G. Nyul [2], raised by the connection between products of quaternions and products of three-dimensional vectors. 1 Quaternionic products and vector products Let H = {r+ x1i+ x2j + x3k | r, x1, x2, x3 ∈ R} be the skew field of quaternions with the basic relations i2 = j2 = k2 = ijk = −1. Throughout this paper we use the following notions for a quaternion h = r+x1i+x2j+x3k ∈ H: We say that h is purely imaginary if r = 0. The conjugate of h is h = r−x1i−x2j−x3k ∈ H, the absolute value of h is |h| = √ r2 + x1 + x 2 2 + x 2 3, and the multiplicative inverse of h is h−1 = 1 |h|2h in case of h 6= 0. Quaternions h1 = r + x1i + x2j + x3k, h2 = s + y1i + y2j + y3k ∈ H are called orthogonal if rs+x1y1+x2y2+x3y3 = 0, or equivalently if h1h2+h2h1 = 0. ∗Research was supported in part by Grant 100339 from the Hungarian Scientific Research Fund.