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Is there a geometric meaning for the notation abuse in 2D?


Whats the derivation for the cross product? Can it be derived from the "More precise definition above"?


How does one get an intuition for the determinant in a way that is not dependent on geometry? This is a vague question, and it's definitely dependent on context, but I've generally gotten intuition for it in the past as a "scaling factor" on volume and I'm just curious to see other ways to make sense of it.