Simple explanation to the RHR?

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Can someone please explain the Right Hand Rule (used in Physics for magnetism), in basic calculus? As we have simply been given a 'vague general rule to use', and no other vector-calculus has actually been taught to us, I have trouble understanding the more "fancy" mathmatical explanations out there, but find the simple "use your hand like this" rules too vague. Hope someone can help!

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RHR comes from the definition of cross product. Because when you define a base to analyse your system, you want the set of vectors to be orthogonal direct (convention + it's convenient for EM). In a direct orthogonal base $(e_x, e_y, e_z)$, $e_x \wedge e_y = e_z$. However, in a indirect orthogonal base, $e_x \wedge e_y = - e_z$. Which is very unconvenient because it would add a bunch of minus signs in EM. Also it is a convention, even for domains of physics that don't use cross product.