Functional equations for vector products and quaternions

Abstract

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.

Pexiderized functional equations for vector products and quaternions

“…As a consequence, we obtain that g : R 3 → H is a solution of functional equation (1) if and only if there exist orthogonal purely imaginary quaternions h 1 , h 2 ∈ H with absolute values 1 such that g ((x 1 , x 2 , x 3 )) = x 1 h 1 + x 2 h 2 + x 3 h 1 h 2 , which is a reformulation of Theorem 1 in [2].…”

“…The author thanks Professor Gyula Maksa, who suggested to study the pexiderized versions of (1) and (2) after reading our paper [2].…”

“…Motivated by these connections between vector products and quaternions, B. Nyul and G. Nyul [2] solved functional equations…”

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AbstractThe 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.

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