The paper demonstrates how electrical variables, when formulated as vector quaternions, can be manipulated. A bottom up approach is taken to analysing the forces on coils in magnetic fields and the derivation of expressions for various scenarios is shown to be simple and quick. Quaternions, which were discovered 170 years ago, are the basis for four dimensional vectors. Electrical variables fit into this realm, as some electrical variables are in quadrature with one another when viewed across different circuit components. The manipulation of the variables to obtain force expressions is shown to be simple and visual when vectors are looked at, in various planes of the four dimensional space for the case of the dc motor. Other applications of quaternions in the field of electrical engineering are also alluded to in the paper and especially 3D vector manipulation, as in spherical motor analysis.
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