Transform as a vector? Tying functional parity with rotation angle of coordinate axes

Abstract

At an undergraduate level, a modern definition of a vector is introduced: a vector is a quantity which remains unchanged under a rotation of coordinate axes. Using this definition, this article determines which of the quantities (triplets) of the form (where x, y and z are coordinate points and ϕ is a real valued function) ‘transform as a vector’ under rotation of coordinate axes, and hence can be designated as a vector. Our approach employs elementary mathematics to determine possible value(s) of rotation of axes angle θ at which C may transform as a vector, even if it does not for all θ. A notable correspondence between the parity of function ϕ and rotation angle(s) is observed. The analysis, initially carried out in an orthogonal coordinate system, is generalized for skew coordinate systems. The intent behind the article is primarily towards providing an improved understanding of the ‘advanced’ definition of a vector.