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Comment by mijoharas | Hacker News Reader

Alright, I got nerd sniped halfway through when he said:

  Scalar triple products only exist in 3D, because they involve cross products which only exist in 3D (please keep the “they exist in 7D too” comments for some other time).

There isn't a reference, and me googling "higher dimensional cross product" seems to return results generalising to any number of dimensions.

Can anyone explain what he's talking about or provide references? Off the top of my head I can't see why you can't generalise cross products to D > 3. Or if you can't I don't see what's special about dimensions 3 and 7...

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Ok... I'll leave the comment to stand but apparently I just needed to scroll a tiny bit more and get an answer here[0].

  The seven-dimensional cross product is one way of generalizing the cross product to other than three dimensions, and it is the only other bilinear product of two vectors that is vector-valued, orthogonal, and has the same magnitude as in the 3D case.[2] In other dimensions there are vector-valued products of three or more vectors that satisfy these conditions, and binary products with bivector results.

[0] https://en.m.wikipedia.org/wiki/Seven-dimensional_cross_prod...

The exterior aka wedge product generalizes, as do determinants. Cross products absolutely do not

Could you explain some more about why?

The simplest counterexample is just to consider vectors in 2D. It is not possible in general to construct a third vector perpendicular to two other vectors unless those other vectors are collinear or anitcollinear. The notion of an exterior product, represented as a signed area subtended by the two vector arguments of the operator sits in 2 space nicely however. In general, signed areas and M dimensional signed volumes are embeddable in N dimensional subspaces for M < N. The notion of a signed volume is directly tied to both the exterior product and the determinant. In higher dimensions, the cross product is equally unhelpful, given that for two vectors, the set of mutually orthogonal vectors abiding by the right hand rule is often infinite.

(should be M <= N but HN isn't letting me edit the comment at this point)

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