Formal statement of the right-hand rule?

[LucasGB]

Is there a way to express the right-hand rule mathematically, without making references to... well, hands?

 

[Landau]

Which right-hand rule exactly are you referring to? That rule concerning cross products?

 

[LucasGB]

Which right-hand rule exactly are you referring to? That rule concerning cross products?

Yes, that's the one.

 

[Landau]

I suppose using the notion of orientation.

 

[UgOOgU]

I think that the right-hand rule is a consequence of the definition of the coordinate system. The three-dimensional euclidian vector space that is usually used in physics is by definition a "right-handed coordinate system". In other words, the versor products of the base 'i x j = k' , 'j x k = i' and 'k x i = j' are defined in this manner. In similar, the versor products in a left-handed coordinate system are defined: 'i x j = -k' , 'j x k = -i' and 'k x i = -j'.

 

[LucasGB]

Apparently, every definition makes use of hands. I wonder if it's possible to define it without referring to that.

 

[LucasGB]

I think that the right-hand rule is a consequence of the definition of the coordinate system. The three-dimensional euclidian vector space that is usually used in physics is by definition a "right-handed coordinate system". In other words, the versor products of the base 'i x j = k' , 'j x k = i' and 'k x i = j' are defined in this manner. In similar, the versor products in a left-handed coordinate system are defined: 'i x j = -k' , 'j x k = -i' and 'k x i = -j'.

Yes, but how is the "right-handed coordinate system" defined without reference to hands?

 

[wofsy]

Apparently, every definition makes use of hands. I wonder if it's possible to define it without referring to that.

You do not need to use hands.

Use the duality between forms and vectors in Euclidean 3 space induced by the standard Euclidean metric.

here are the steps:

1) to each vector v associate its dual linear function, <v,>.

2)Take the wedge product of these two dual forms. The wedge product is the oriented area element of the paralleogram spanned by the two vectors.

3)Take the dual vector of the wedge product. This is the cross product.

 

[UgOOgU]

Yes, but how is the "right-handed coordinate system" defined without reference to hands?

How? In the manner that I have defined in my prior post. In there I had established the cross products of the base of the space, it has not reference with hands or with the "common representation" of these base.

 

[LCKurtz]

Suppose you are in radio contact with an alien who "looks like us" in that he has an arm with a hand on each side of his body. You are trying to explain to him which side is his right side. I don't think it is possible unless there is some sort of asymmetric experiment you can ask him to perform, which I doubt. Even if he understood 3D and three mutually perpendicular axes, you couldn't explain which orientation is "right handed" either.

I'm willing to be wrong about this though

 

[Phrak]

You posted this in a mathematics folder, rather than a physics folder. If you had posted in physics we would refer handedness to measurable things like handedness in relation to the fingers on our hands, or the statistics of the weak particle decay.

But in pure mathematics there are no physically measurable things. There is no reference to right-handed vs left-handed. They are simply duals of one another. We can call the labels anything we want. Left and Right are just labels to distinguish one from the other.

 

[Landau]

Apparently, every definition makes use of hands. I wonder if it's possible to define it without referring to that.

As I said, not using the concept of oerientation. See wofsy's post.