how to determine the direction of cross product
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For our commonly perceived 3D world, it is perpendicular to both vectors. Its direction is given by the right hand rule. Right-hand rule - Wikipedia
If you live in 2 dimensions, it is a (signed) scaler. If you live in more than 3 dimensions, say D, then the “cross product” (i.e. exterior product) is a D-2 form called the “Hodge Dual” and is specified by the Hodge star operator. But that may getting a little weird and esoteric. You are probably just looking the answer given in the first paragraph.
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