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I have used cross products thousands of time without really knowing what it actually does; I know how to compute it, but I don't feel like I understand it. Also, when it shows up in physics/kinematics contexts, it's only because the magnitudes of the vectors involved have to be multiplied and with the sine of the angle between them (and have to point in a direction perpendicular to both vectors). I know this is equal to [itex]X\times Y[/itex], but I don't know quite why.

I am looking for one or more resources which can explain it both rigorously and intuitively (not necessarily simultaneously).