Vectors and Quaternions

Abstract

PROF. MACFARLANE claims that his “fundamental rules for vectors are based on physical considerations, the principal one of which is that the square of a vector is essentially positive.” His proof is virtually this:—The expression for the kinetic energy (½ mv²) is an essentially positive quantity. It contains one factor ½m evidently positive. Hence the other factor v² must also be positive. “But v denotes the velocity which is a directed quantity.” Unfortunately for this argument v does not denote the velocity in its complete conception—it simply measures the speed. The physicist may think of velocity as being a vector quantity; but in ordinary analysis the vector is not symbolised. We deal only with tensors and scalars. It would be well, I think, if the strict meaning of vector were clearly borne in mind. A vector is a directed line in space, and may be used to symbolise all physical quantities which can be compounded according to the well-known parallelogram law. Displacement is perhaps the simplest conception that can be so symbolised. Velocities, concurrent forces, couples, &c, are in the same sense vector quantities. Now it can be proved rigorously that quadrantal versors are compounded according to this very addition law. On what grounds, then, are they refused admittance to the order of vectors? If a vector cannot be a versor in product combinations, what is the significance of the equation ij = k? Regarding this Dr. Macfarlane vouchsafes no remark, save that it is possible to get along without its use. As he himself has not done so, such a possibility lies altogether outside our consideration. Again, I fail to see what “physical considerations” have to do with mathematics of the fourth dimension.