>"untyped"

It's amazing because that's the diametric opposite of what I'd consider to be true this is what the non-GA approach does, and which we avoid!

The modern approach to GA, eg https://en.wikipedia.org/wiki/Plane-based_geometric_algebra, distinguishes planes-through-the-origin, points, lines-through-the-origin, and lines-at-infinity. These four different types all get called "vectors" in the ordinary mindset. The cross product is said to take in two "vectors" and give you a "vector", but in reality it's generally taking two planes and giving you a line. This is talked about in the fourth paragraph of the wiki article :)