This is incorrect, if you read Spivak, you can define and (n-1)-ary product which generalizes the cross product. You give it (n-1) vectors and it gives you a vector orthogonal to all of them.
Whether you call it the cross product or not is just semantics, but it does exist in terms of the exterior product.
It's difficult to find online, but it's constructed directly in Spivak. A quote:
"It is uncommon in mathematics to have a "product" that depends on more than two factors. In the case of two vectors v,w in R^3, we obtain a more conventional looking product, v X w in R^3. For this reason it is sometimes maintained that the cross product. can be defined only on R^3" - Calculus on Manifolds, pg.84
Most graduate students read this (or did five years ago).
> This is incorrect, if you read Spivak, you can define and (n-1)-ary product which generalizes the cross product. You give it (n-1) vectors and it gives you a vector orthogonal to all of them.
I never said the wedge was the only way to generalize the cross product, I just said the cross product itself wasn't general.
It's just semantics: what you call the generalized cross product, I call the cross product.
We used to think as negative numbers being a generalization of the integers, so that's some food for thought. I'm sure once quantum mechanics dominates solving eigenvalue problems will be a high-school level problem, so we'll end up having complex numbers losing their complexity. We don't call them "real" numbers outside of math circles anymore.
To say that the cross product itself is a hack is a bit of a stretch though, it can easily be generalized and I think it's quite natural.
These words all change meaning over time, mathematical definitions change, old words are used to describe new objects and new words are used to describe old objects. I'm using the word integer with it's archaic meaning here for rhetorical effect, but I'm probably being too obtuse ;)
To clarify, we used to think of integers as just the natural numbers. Integer was a colloquial word meaning "whole, entire", so there was presumably a discussion about how negative numbers were whole or entire, though I vaguely remember this historical story. My point is just that at some point negative numbers were seen as a advanced extension of the natural numbers. Even zero was seen as an unnatural extension, which is why there is a confusion to this date as to whether "natural" numbers include zero.
Then again, with these conversations, the N < Z < Q < R < C... classification automatically pops in your mind if you did mathematics in last classes of high school (that's going to be a LOT of laypeople ! Of course, a lot of them, those that rarely encounter them, might then forget about this classification.)
And integers are still called "natural integers", and negative numbers are NOT called "natural" ?
Whether you call it the cross product or not is just semantics, but it does exist in terms of the exterior product.
It's difficult to find online, but it's constructed directly in Spivak. A quote:
"It is uncommon in mathematics to have a "product" that depends on more than two factors. In the case of two vectors v,w in R^3, we obtain a more conventional looking product, v X w in R^3. For this reason it is sometimes maintained that the cross product. can be defined only on R^3" - Calculus on Manifolds, pg.84
Most graduate students read this (or did five years ago).