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I understand how to calculate a cross product, but why for instance is the cross products of two vectors another vector that is perpendicular to it. Can you prove or explain this to me in anyway. There are two parts I want you to answer here:

1. How this calculation yields another vector

2. Why this vector is perpendicular to the other two vectors.

After i have understood the 2-part question above. Why is the cross product of the unit vectors i x j=k and j x i= -k? After all the answer of a cross product is vector that is perpendicular to the other vectors. In the example i x j = k, the vector -k is just as perpendicular to these two vectors as the vector k, why can't the answer of i x j be equal to also -k. In the same way why is j x i= -k, the vector k is just as perpendicular as -k, so why can't j x i be equal to vector k. If this is just convention that scientist agree upon, then i don't understand how we can expect nature to "conform" to standard conventions that everybody has agreed upon.