A 1-vector is just a regular vector. A 2-vector (or “bivector”) is a quantity associated with a two-dimensional “direction”, which is an oriented plane. And so on.
Ok, but how do you actually “and so on” the orientability here? I have not actually tried to picture how you orient a 3-vector in a higher space. And I’m suspicious about my analogy between 1-vector and 2-vector orientation until I can picture that. (You can orient a plane by picking one of the two halves it divides a 3-d volume into, but you normally orient a line by thinking about the ends, not the sides where it divides the plane. Does that matter?)
I think the way this all works is a lot more subtle than I’ve been imagining, and probably some of the stuff in the original shortform about orientation is wrong.
Ok, but how do you actually “and so on” the orientability here? I have not actually tried to picture how you orient a 3-vector in a higher space. And I’m suspicious about my analogy between 1-vector and 2-vector orientation until I can picture that. (You can orient a plane by picking one of the two halves it divides a 3-d volume into, but you normally orient a line by thinking about the ends, not the sides where it divides the plane. Does that matter?)
I think the way this all works is a lot more subtle than I’ve been imagining, and probably some of the stuff in the original shortform about orientation is wrong.