Why is the cross product is alternating?

In the book An introduction to manifolds by Tu, Loring W, it says that

The cross product $$v\times w$$ on $$\mathbb R^3$$ is alternating.

However in the definition of alternating, it says that

A $$k$$-linear function $$f:V^k\to\mathbb R$$ is symmetric, if \begin{align*} f\left(v_{\sigma(1)}, \ldots, v_{\sigma(k)}\right)=f\left(v_{1}, \ldots, v_{k}\right) \end{align*} for all permutations $$\sigma\in S_k$$; it is alternating, if \begin{align*} f\left(v_{\sigma(1)}, \ldots, v_{\sigma(k)}\right)=(\operatorname{sgn} \sigma) f\left(v_{1}, \ldots, v_{k}\right) \end{align*} for all $$\sigma\in S_k$$.

However, I can't understand why the cross product is alternating. I think that the cross product $$\times$$ can be viewed as a function $$\mathbb R^3\times \mathbb R^3\to\mathbb R^3$$, but not any function that maps to $$\mathbb R$$. So why this is alternating?

You're correct that using the given definition strictly, this doesn't make sense. However, there are a couple small changes in interpretation you could choose to make it work:

1. Extend the definition of a symmetric and alternating functions to apply to $$k-$$linear functions $$V^k \to W$$ for any vector space $$W$$. (You can use the same defining formulae.) In the case of the cross product, it's alternating in this sense simply because $$u \times v = - v \times u.$$

2. "Lower an index" on the cross product via the metric, i.e. consider $$f(u,v,w) = (u \times v) \cdot w.$$ This is then a 3-tensor on $$\mathbb R^3$$ that co-incides with the determinant.

• Could you please expand on via the metric? Except from dot product, what can also be used to "Lower an index"? Nov 24 '20 at 1:41
• @FFjet: in Riemannian geometry there's a generalization of the dot product called the Riemannian metric, and it is quite common to implicitly use it to change the type of a tensor. This is also known raising/lowering indices or as the musical isomorphisms. If you're just beginning to learn differential geometry then I wouldn't worry too much about that (and I'm surprised Tu's book seems to assume you're familiar with the idea). Nov 24 '20 at 1:45
• Ok, thanks a lot :) Nov 24 '20 at 1:48

If you expand the definition of alternating to linear functions from $$\mathbb{R}^n$$ to $$\mathbb{R}^m$$, one has the following.

If $$u$$ and $$v$$ are linearly independant vectors, then $$u\times v$$ is defined to be the unique vector such that

• $$\|u\times v\| = \|u\|\|v\||\sin(u,v)|$$
• $$u\times v$$ is orthogonal to the plane generated by $$u$$ and $$v$$
• $$(u,v,u\times v)$$ is a positively oriented basis

Thus, if $$(u,v,u\times v)$$ is positively oriented, then $$(v,u,u\times v)$$ is negatively oriented. Hence, $$(v,u,-(u\times v))$$ is again positively oriented. The norm of $$-(u\times v)$$ is the good one, and it is orthogonal to $$u$$ and $$v$$. By uniqueness, $$-(u\times v) = v\times v$$.

As the cross product is bilinear, and the only permutations on a set of two elements are the identity or the transposition, it follows that the cross product is alternating.