On bivectors and jay-vectors

M. Hayes, N. H. Scott

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Abstract

A combination a+ib where i2=−1 and a, 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 a+jb of real vectors a and b, where j2=+1 but j is not a real number, so j≠±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.
Original languageEnglish
Pages (from-to)859-882
Number of pages24
JournalRicerche di Matematica
Volume68
Issue number2
Early online date6 Apr 2019
DOIs
Publication statusPublished - Dec 2019

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