The better idea is to start with vectors. Adding vectors or scaling them is straight-forward. It’s trickier to figure out what it means to multiply or divide two vectors.
The quotient of two vectors v/u should be some kind of operator which transforms one into the other (that is, when you multiply it by one, you get the other, (v/u)u = v(u\u) = v, because we want multiplication to be associative). If those two vectors are the same length but different directions, the natural transformation to use is a rotation. The reason to use a rotation is that we want the transformation to make sense irrespective of any arbitrary coordinate system we decide to impose. If we used some kind of skew, it would break down under change of coordinates. As for reflections: if the quotient of two vectors was some kind of reflection, then we could square any quotient of vectors to get an identity transformation, which would not result in a very useful or consistent arithmetic.
The quotient of two vectors v/u should be some kind of operator which transforms one into the other (that is, when you multiply it by one, you get the other, (v/u)u = v(u\u) = v, because we want multiplication to be associative). If those two vectors are the same length but different directions, the natural transformation to use is a rotation. The reason to use a rotation is that we want the transformation to make sense irrespective of any arbitrary coordinate system we decide to impose. If we used some kind of skew, it would break down under change of coordinates. As for reflections: if the quotient of two vectors was some kind of reflection, then we could square any quotient of vectors to get an identity transformation, which would not result in a very useful or consistent arithmetic.
The nicest and most useful formalism for defining multiplication of vectors is called geometric algebra, a.k.a. Clifford algebra. Start with http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf
Or see the recent blog post http://www.shapeoperator.com/2016/12/12/sunset-geometry/
Or see more links at https://news.ycombinator.com/item?id=12938727#12941658