In definition of vector multiplication, the direction of the resulting vector is given by the right hand rule. However I don't know any mathematical requirement to pick right hand instead of left for this purpose. Is there any?
You could define it with the left hand, too. In this case, the formula of the 3d vectorial multiplication would simply negated.
A physics could be also constructed for that, it would be exactly the same, of course the formulas using vectorial multiplication would be negated.
It is like a binary Higgs-mechanism, similarly as the analog, mechanical clocks are rotating to right. They could rotate also left. Some hundreds of years ago, a lot of them did.
Draw a 3D set of axis for XYZ. Where is positive X? Where is positive Y? Where is positive Z? Obviously wherever you want them but right, up and out of the page are the commonly accepted choices.
Now accepting that how do I remember it? I close the fingers of my right hand from X to Y and my thumb points to Z. If I instead preferred opening my fingers from X to Y I would use need to use my left hand.
Also this lets teachers be pretty sure who is going to fail during tests.
EDIT: The relation of the orientations is part of the definition of a cross product. We want to avoid adding negative signs if we don't have to. Using the traditional XYZ this means closing your right hand, but each time you flip the sign of an axis you can add a switch of hands to preserve a "stay positive" property.