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I guess I have a few questions.

1)How did mathematicians or physicists discover how to calculate the determinant of a 3 x 3 matrix, and what does this determinant represent?

For example, A x B = (AyBz - ByAz)i - (AxBz-BxAz)j + (AxBy-BxAy)k.

2) How did we know to put a negative in front of the j term?

3) How did we recognize that phenomena like Torque and Magnetic Force could be described by cross products?

I hope that I am being clear enough and don't sound like a confused idiot. I'd appreciate answers to any of these questions.