On bivectors and jay-vectors

  • M. Hayes
  • N. H. ScottEmail author


A combination \(\varvec{a}+{\mathrm{i}}\varvec{b}\) where \({\mathrm{i}}^2=-1\) and \(\varvec{a},\,\varvec{b}\) 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 \(\varvec{a}+{\mathrm{j}}\varvec{b}\) of real vectors \(\varvec{a}\) and \(\varvec{b}\), where \({\mathrm{j}}^2=+1\) but \({\mathrm{j}}\) is not a real number, so \({\mathrm{j}}\ne \pm 1\). 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 is discussed briefly. Application is made to second order elliptic and hyperbolic partial differential equations.


Split complex numbers Hyperbolic numbers Coquaternions Conjugate semi-diameters Hyperboloids and ellipsoids PDEs 



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© Università degli Studi di Napoli "Federico II" 2019

Authors and Affiliations

  1. 1.School of Mechanical and Materials EngineeringUniversity CollegeDublinIreland
  2. 2.School of MathematicsUniversity of East AngliaNorwichUK

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