On bivectors and jay-vectors

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.