pconstantino said:
Do you know how this was created?
I think the point you keep missing is that things like the vector cross product are invented, not discovered. And they are invented specifically to correspond to nature.
Vectors are a fairly new concept in math. They are not a fundamental part of nature, they are just constructs people invented to help describe and predict what happens in nature.
So we can define operations among vectors any way we want.
Start with addition. When you add two vectors, you can just add their components. That seems like a natural way to do things, and it turns out to be useful. So that's how we define vector addition, and everybody's happy.
Now how about vector multiplication? Again, the natural thing might be to just multiply their components, but if you do that, you don't get anything very useful.
But if you multiply the components and then add them together to get a scalar, that does turn out to be useful in describing things like work, so we agree to call that the dot product.
But then somebody else sees that if we multiply the lengths by the sine of the angle, we get another useful quantity. And if we also stipulate a direction perpendicular to the two vectors, we get both the quantity and direction of certain things we find in nature, like torque. So we DEFINE a second way to multiply vectors, and give it a different name, namely the cross product.
It's not that torque or magnetic fields know anything about the cross product; they just do whatever they do. But we notice that they obey some laws we can discover --- we may or may not understand why they obey those laws, but we can observe that they do --- and we INVENT the cross product to help us describe what is happening.
If torque or magnetism followed a different law, then we would have to invent a different operation to describe it. It's not magic that torque follows the cross product, any more than it is magic that a rock falls "down" when you drop it. We just invented "down" to describe the direction we see rocks fall. At a very fundamental level, nobody knows why they fall; they just do. We can observe and measure and predict what will happen with great precision, but we don't really know why gravity does what it does.
So we just invented the cross product to describe what we see happening in nature. And like everybody else is trying to tell you, whether we define the perpendicular direction as right hand or left hand is not important; we would just have to change the sign if we changed hands. What is important is that we pick one, and everybody agrees on it, so we can communicate.
That's really all there is to it.