[tobinfricke]

The cross product is a peculiar animal, as it only exists in 3-dimensional space. There is no unique "cross product" in 4 dimensions or higher. (In two dimensions we cheat and define the result of the cross product as the scalar magnitude of what would be the component in the third dimension, if it existed.) Furthermore, it turns out that the result of taking the cross product of two vectors is itself not a vector. In physics we interpret a vector not simply as "an ordered list of numbers," but as a geometric quantity that responds to coordinate system changes in the expected way. Suppose that we have two vectors, a and b, and we take their cross product. Now suppose we choose to work in the "mirror image" coordinate system. Our choice of coordinate system should not affect physical outcomes. But while "a" and "b" are inverted in our mirror image coordinate system, the cross product "a x b" does NOT invert.

Introductory physics textbooks proceed to tell us that "well actually," the result of a cross product (such as an angular velocity vector or the Poynting vector of electromagnetism) is actually a "psuedovector."

Other formalisms treat the cross product in a more hygienic and general manner.

This is all to say that familiar mechanisms like the "dot product" and "cross product" are not necessarily as "natural" as you may have been lead to believe.

[paholg]

There's also a cross-product in 7 dimensions, but just those two: https://en.m.wikipedia.org/wiki/Seven-dimensional_cross_prod...

I wish we would stop teaching/using the cross-product. Bivectors make a ton more sense, and as a bonus do away with the right hand rule.

[creata]

> the "dot product" and "cross product" are not necessarily as "natural" as you may have been lead to believe

The cross product, sure: its problem is that it dualizes unnecessarily, making you deal with a normal vector when you almost always just want the plane.

But what did the dot product ever do to you?

[tobinfricke]

> But what did the dot product ever do to you?

Fair enough.

What I was getting at is that "standard vector analysis" is a choice, and it turns out that there are alternatives where things are defined differently.

[npsomaratna]

But don't you need the normal vector to define the plane? (Genuine question)

[alok-g]

A bivector is just as good for the purpose. Think of it as a generalization of a directional arrow (vector) into an oriented area (bivector). The OP talk shows good visualizations if you haven't watched yet.

[npsomaratna]

Got it, thank you. (And facepalm, this is something I should have remembered!)

[tobinfricke]

A plane does not have a unique "normal vector" in higher than 3 dimensions.

[npsomaratna]

Good point, thank you.

[pandaman]

If you define cross product through minors, then you can expand it to any dimension, just with the different number of arguments, n-1 for R^n, 1 in R^2: [{x,y},] = {y,-x} for example. 3 in R^4, etc. Going to have the same properties as R^3 wrt to linearity and anticommutativity.

[tobinfricke]

I think you are basically reinventing the exterior / wedge product, which is indeed the generalization of the cross product.