Somehow, these topics often seem to devolve into discussions of how cross products are evil and (insert alternative) is better or into math full of jargon that’s incomprehensible without studying the math in question. [0]

I’ll try to say something intelligent about this identity. This comes from a class I took on different geometry, but I’ll leave out the differential geometry :)

First, u, v, w, etc are n-dimensional vectors [1]. n = 3 here, and this is important later.

The object [u v w] is a function of three vectors. You feed it three vectors and it spits out a number. It has two interesting properties:

1. It’s multilinear. This means that [u+v w x] = [u w x] + [v w x] and [a·u w x] = a·[u w x]. The same holds for the other two parameter. This just means that, if you fix all but one parameter, you’re left with a linear function of the parameter you didn’t fix.

2. It’s totally antisymmetric. If you swap any two parameters, you negate the result. So [u v w] = -[u w v] = [v w u]. Play with this — it’s fun. With three parameters, the cyclic permutations are positive. With any number of parameters, the even permissions are positive.

These properties are pretty easy to prove from the definition of the triple product.

Now on to the messy function in the article:

f(u,v,w,x) = u[v w x] – v[w x u] + w[x u v] – x[u v w]

f is multilinear: each term is fairly trivially multilinear, and the sum of multilinear functions is multilinear.

f is antisymmetric. Try it — swapping any two arguments negates it! [2]

Now for the fun: In n dimensions, if you have a k-parameter antisymmetric multilinear function, then, if k>n, your function is always zero!

Lots of other cases are interesting, too:

k=1: these are just linear functions of vectors. One might call them “dual vectors” or “covectors” or whatever. If you are a bit sloppy, you can thing of them as vectors, and they are vectors, but they are not the same type of vector as the type of their input. If you cough erase that type difference, you end up in the rabbit hole that leads to cross products making sense in 3D but not otherwise.

k=n: This is a “volume element”. In high school calculus, x·dx or f(x,y)·dx·dy are things that live inside an integral and you aren’t supposed to do too much in the way of removing the integral sign. In 1D, x·dx is a function mapping x (a point in space) to dx, a volume element. If you have a mapping from points to volume elements, you can integrate it and get a scalar! In Euclidean space, you can get away with integrating something like a volume element (e.g. dx·dy) and ignoring the point part. In non-Euclidean space, a vector here and a vector there are different things — a vector (say, the direction and ant is crawling if it’s at a certain spot on a balloon) is not a vector elsewhere* — an ant crawling that direction elsewhere on the balloon whirl be crawling off the surface!

k=0: scalar. It’s a scalar-valued function of zero parameters. That would be a scalar.

Other goodies from high school geometry and calculus are hiding in here, too, with appropriate k and n.

[0] For example, “Algebra” means something to most people who went to high school. It means something rather different in fancy math. I don’t know the history of how this came to be.

[1] If you are working with a non-Euclidian manifold, then they’re all vectors originating at the same point. Yes, this all generalizes to any differentiable manifold.

[2] If you feel fancy, the “mess” is the exterior product of the identity on one vector with the scalar triple product, times plus or minus 1 (I didn’t check). And wow, the Wikipedia article on the exterior product is a mess. You can read several pages, try to remember a bunch of math you haven’t used in a while, and still have no idea how to compute the thing. It’s really not that bad once you get past the jargon.