I think it is a good idea because both components contain useful complementary information (the cosine/sine of the angle and the plane they form), and the algeb

I think it is a good idea because both components contain useful complementary information (the cosine/sine of the angle and the plane they form), and the algebra corresponds to geometric operations. Having an algebra that corresponds to geometric operations is good because you get a high-level framework for how to manipulate quantities instead of having to do things case by case.