Spherical-vectors and geometric interpretation of unit quaternions

Abstract

The purpose of this work is to introduce and study the notion of spherical-vectors, which we can consider as a natural generalization of the arguments of complex numbers in the case of quaternions. After having established some elementary properties of these particular vectors, we show by transport of structure that spherical-vectors form a non-abelian additive group, isomorphic to the group of unit quaternions. This identification allows us, first, to present a new polar form of quaternions, then to represent the unit quaternions on the unit sphere of $\mathbb{R}^3$ and to interpret their multiplications geometrically.Comment: 20 pages, 9 figure