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> 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?

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> 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.

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

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.

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

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

Good point, thank you.

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