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What's up with all the foo-vectors?

This is an attempt to succinctly (hah) answer a question I keep having to refresh my memory about: What's up with (vectors, bivectors, axial vectors / pseudovectors, multivectors, the cross product, etc.?) How do they relate to each other?

Multivectors

Multivectors or k-vectors are a generalization of vectors. Vectors have a length and a direction, and can be thought of as one-dimensional; k-vectors generalize vectors to arbitrary dimension k. In this framework, a scalar -- a quantity without direction -- can be thought of as a 0-vector. 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.

What does it mean for a plane to be "oriented"? It means we pick one side to be the "right side" and the other to be the "wrong side". (In the same way, a vector is an "oriented line", which has a "right end" where we draw the arrowhead.)

The exterior ("wedge") product

We get multivectors from vectors using the exterior product, or "wedge product". In the 3-dimensional setting, the wedge product smells almost exactly like the cross product -- it takes two vectors and gives back a bivector, whose magnitude is the area of a parallelogram formed by those two vectors, and whose orientation depends on the relative directions of the two vectors. (I'm being deliberately vague here to avoid saying anything false; I could say "according to the right-hand rule" to get the general point across, but a later point will be that the left-right choice here is arbitrary, and could have been chosen the other way.)

Pseudovectors

In an n-dimensional space, a pseudovector (or axial vector) is an (n-1)-vector -- that is, an n-minus-one-dimensional multivector. (A pseudoscalar is an n-vector.) Consider a 3-dimensional space: A bivector picks out two dimensions of it (an oriented plane), but picking two out of three dimensions leaves just one dimension remaining un-picked. So every bivector (a plane with magnitude and orientation) can be matched up with some vector (with the same magnitude, and pointing normal to the plane in the direction of its orientation.)

So in 3-dimensional space, a bivector is a pseudovector, because it is very nearly equivalent to a regular vector. (And a trivector is a pseudoscalar -- there is only one possible basis-trivector, since there are only three dimensions and it has to span all of them. So a pseudoscalar only has a magnitude, and no meaningful direction, just like a regular scalar.)

Orientation

Why did I say "very nearly equivalent" -- what's the "pseudo" part about? This is trickier to explain, and while it will work fine as a refresher for myself, I don't know if I will get it across fully to anybody else, but I'll try.

Consider unit vectors pointing along the X, Y, and Z axes. Consider also a bivector X ^ Y, which has unit magnitude, and is oriented with the "right side" pointing in the same direction as our Z vector.

Now, flip the whole space around as though you are looking at it in a mirror. You can do this by e.g. negating any of our three vectors. In the resulting space, the X ^ Y bivector is now oriented in the opposite direction from the Z vector. (Thinking about the right-hand rule, consider that a right hand viewed in the mirror looks like a left hand. So if we apply the right-hand rule to the mirrored space, it will point in the opposite direction from how it pointed in the non-mirrored space.) If you take the "pseudovector" view of it -- treating X ^ Y as something like a vector pointing along the Z axis, instead of a plane oriented towards the +Z axis -- you will see where the "psuedo" comes from. Reflecting the space in a mirror causes the vector and the pseudovector, which pointed in the same direction before, to now point in opposite directions.

If you haven't encountered this before, it's probably going to seem like sophistry or handwaving, sorry. All I can say to that is, I promise this actually makes a difference, although I cannot adequately explain why at this time.

The cross product

This all comes around to why people say things like "the cross product doesn't really give a vector!" Because if you look at the universe in a mirror, the result of the cross product does not behave like a vector. It will not appear mirrored like regular vectors, because its direction depends on handedness, and mirrors reverse handedness.

This also explains why sometimes people say "the cross product gives a bivector" and other people say "the cross product gives a pseudovector". In 3-dimensional space, which is the only place the cross product is well-defined, the two are equivalent.

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?)