Vector Expression of Units of Physical Quantities

Abstract

Vector representations of physical quantities in pseudo-Euclidean space are used to discuss several options for an appropriate coordinate system used to express a system of units. The SI system is found to have several issues causing the system of units to be overdetermined and underdetermined. We propose a version of the SI system that takes into account the three-dimensional nature of physical space and eliminates “dimensional isotopes,” i.e., quantities with different physical meaning but identical dimensions in the SI system. We demonstrate that the mathematical expression for the vector norm of a measurement unit, referred to as a “system unit,” can be used to determine whether a unit is within the range of variation in such a system unit.