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You're in for a nice trip, the concept is called Geometric Algebra:

https://youtu.be/60z_hpEAtD8?si=HHs_9m0IJ43nfI3S (~50m video)

TLDW: Yes, the concept is there, makes much more sense than a cross product (which is just an oriented area) and generalizes really nicely.

Alternatively, read: https://en.m.wikipedia.org/wiki/Bivector

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Gotcha; it's less that my complaint doesn't apply, but more that it isn't relevant (i.e., "squares" and "cubes" aren't especially interesting constructs which need to coexist nicely, and if you relax that constraint then directional geometry can be very interesting). Does that sound right?

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