On bivectors and jay-vectors

Abstract

A combination $\mathbf{a}+\mathrm{i}\mathbf{b}$ where ${\mathrm i}^2=-1$ and $\mathbf{a}, \, \mathbf{b}$ are real vectors is called a bivector. Gibbs developed a theory of bivectors, in which he associated an ellipse with each bivector. He obtained results relating pairs of conjugate semi-diameters and in particular considered the implications of the scalar product of two bivectors being zero. This paper is an attempt to develop a similar formulation for hyperbolas by the use of jay-vectors - a jay-vector is a linear combination $\mathbf{a}+\mathrm{j}\mathbf{b}$ of real vectors $\mathbf{a}$ and $\mathbf{b}$, where ${\mathrm j}^2=+1$ but ${\mathrm j}$ is not a real number, so ${\mathrm j}\neq\pm1$. The implications of the vanishing of the scalar product of two jay-vectors is also considered. We show how to generate a triple of conjugate semi-diameters of an ellipsoid from any orthonormal triad. We also see how to generate in a similar manner a triple of conjugate semi-diameters of a hyperboloid and its conjugate hyperboloid. The role of complex rotations (complex orthogonal matrices) is discussed briefly. Application is made to second order elliptic and hyperbolic partial differential equations. Keywords: Split complex numbers, Hyperbolic numbers, Coquaternions, Conjugate semi-diameters, Hyperboloids and ellipsoids, Complex rotations, PDEs MSC (2010) 35J05, 35L10, 74J05