But why would the area be expressed as a length? This always struck me as a coincidence, and having read on bivectors it just makes sense that it would be (because a vector represents a distance on a line, a bivector represents an area on plane, a trivector represents a volume on a 3D space and so on).
Area is not expressed as a length. In general you can't simplify an expression that is a sum of an area and a length (this is one of the features of geometry that GA handles for you algebraically). e.g. "one meter plus one meter" can be simplified to "two meters", but "one square meter plus one meter" cannot be simplified. However if one of the terms is zero you can remove it, so "zero square meters plus one meter" is "one meter".
Right, with bivectors as in the article it all works.
What I was saying is that summing areas and length is exactly what happens with vector product. The k vector (or the k imaginary unit in quaternions) is the third unit vector, so it has length 1. But when I compute i⨯j=k, I suddenly interpret it as the area of the parallelogram formed by a and j, and at the same time k is the direction perpendicular to both i and j so its coefficient must be a length.
Likewise for triple product which computes a volume but it expresses it as a number (i.e. a length). We study all of these things in vector calculus and don't pay attention to these inconsistencies, but they are there.
It's even more confusing because the triple product is actually a signed volume (pseudoscalar). I admit that I also never noticed these problems until learning GA.
