Vectors as quaternions: a corner of linear algebra

Extract

A recurring topic of discussion is the perceived difficulty, or awkwardness, of establishing the fundamental rules of vector algebra by rigorous geometrical arguments (see, for example, [1]). The following exposition is an attempt to show that, for those acquainted with the rudiments of linear algebra, the essentially simple theory of vector algebra can be clarified by presentation in the context of quaternion algebra, whose elegance and usefulness may still be somewhat undervalued. We make no explicit use of trigonometry, but of course some underlying geometrical ideas are mentioned.