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AFAICT you can do all of physics without cross products....
IMHO it's easier and better without cross products.
Use /wedge products/ instead.
The cross product is defined in terms of a “right hand rule”.If I understand correctly, the wedge product a^b of 2 vectors a, b is synonym of their outer product as defined in geometric algebra (see, e.g., Fig. 2 in [1]). In this case cross product does differ from wedge/outer product for spaces with D>3, simply because for D>3 there are more than one lines perpendicular to plane (a, b). But if restricted to D=3, the difference seems to disappear - either definition specifies its outcome uniquely. In my view, the statement that a "...wedge product is defined without any notion of handedness, without any notion of chirality" results from subtle substitution of terms. The fact that by-vector a^b lives in (a, b)-plane does not eliminate handedness. Just as the line perpendicular to (a, b) may have two opposite directions, the parallelogram representing by-vector a^b is a 2-sided plane segment. Calling the opposite sides "black" and "white" respectively, we have to decide which one of them will face, e.g., the upper semispace (see again Fig. 2 of the same reference). Making such choice would be equivalent to choosing the "right or left hand rule" in the cross product. Reference [1]: D. Hestenes, Am. J. Phys., 71 (2), 2003
A wedge product is defined without any notion of handedness,
without any notion of chirality.