The right hand rule is a common convention for describing orientation of coordinates, used throughout physics. It's also used in the definition of the cross product.
Is it possible to give a purely mathematical definition of the right hand rule, which doesn't reference anyone's hand? What is unique about the right hand that the RHR captures?
If there is no mathematical way to do this, other than by reference to a known object: please explain why this can't be done. Is it similar to e.g. a unit length, which can only be defined by reference to a known object? How does the cross product, which seems to be purely mathematical, depend on it?
Update
This paper https://sites.math.washington.edu//~marshall/math_136/right-hand-rule.pdf seems to state that a version of RHR can be specified using only determinants:
Theorem. A vector triple (u, v, w) satisfies the relaxed right-hand rule if and only if it has a positive determinant
However, without reading it carefully, I'm not sure if it really achieves independence from anatomy, or merely obscures it. It defines:
Given three vectors u, v and w that are not coplanar, we say that the vector triple (u, v, w) satisfies the relaxed right hand rule if n(u, v) forms an acute angle with w
which seems to beg the question which of the 2 angles formed we look at to see if they're acute? Or: which of the two directions along $n$ do we look at to see if the angle formed is acute?
So I'm not sure if the paper is solving the problem or merely obscuring it.
Update 2
Gerry Myerson writes:
It would help if you would give a precise statement of what you mean by "the right hand rule".
Others point out that any right hand rule can be changed to a left hand rule. The best write up for that is here:
The right hand rule is arbitrary - a left hand rule would have worked equally well... it is always the case that these sort of right/left hand rules occur in pairs that make the observable result not arbitrary
Hey, using my left-hand for right-hand-rule problems (with electrons, which have negative charge) was a lifesaver during exam-time: I could keep writing with my right hand while orienting my left hand to match the physical diagram of the system.
Yet, vector calculus texts do indeed use the RHR:
If $e_1, e_2, e_3$ satisfy the RHR, then $\det[a, b, c] > 0$ if $a,b,c$ (in that order) satisfy the RHR.
-- Hubbards, Vector Calculus, Section 1.4 [abridged] (Hubbards defines the RHR by drawing pictures of hands.)
Note that the Hubbards definition meets the claim above, that RHR always occur in pairs, and a LHR could be used as well. (Similar perhaps to $i$ being indistinguishable from $-i$.) Nonetheless, we can still define $i$, and should still be able to define RHR, by its properties. Here's my attempt, to which I request feedback:
Consider three linearly independent vectors $a, b, c \in \mathbb R^3$. There is a unique rotation $\rho$ that meets the following conditions:
- $\rho(a) = pe_1, p > 0$
- $\rho(b) = qe_1 + re_2, r > 0$
- $\rho(c) = se_1 + te_2 + ue_3$
If $u > 0$, then $a, b, c$ is said to meet the RHR.
I believe the above is a correct definition that is mathematically rigorous, self-contained, requires no anatomy, and still captures the way that RHR is used in math (at least according to Hubbards and the other texts I've examined) and physics.
Essentially, this is a formal way of defining, for $\mathbb R^n$ if $n$ linearly independent vectors are in the "same orientation" as a given basis (implicitly the standard basis). This suggests a broader question: In $\mathbb R^n$, two sets of $n$ linearly independent vectors can be in the same orientation or opposite orientation; the above is a test for $\mathbb R^3$, which can be extended to any $n$.
Am I correct? Can you help me prove the existence and uniqueness of $\rho$? And that this definition of RHR fits Hubbards' claim regarding the determinant?
Note: I've formalized my conjecture in this post, where I present its implications and ask for help proving it: Prove this conjecture: Two lists of vectors are in the same orientation iff this transformation exists
I think the issue you are trying to resolve is the one where there is simply not a mathematical reason for choosing one orientation over the other. "Orientation" is the fact that there is a natural equivalence relation on the set of ordered bases of $\mathbb R^3$: if $(e_1,e_2,e_3)$ and $(f_1,f_2,f_3)$ are ordered bases of $\mathbb R^3$, then there is a unique linear map $\alpha\colon \mathbb R^3\to\mathbb R^3$ such that $\alpha(e_i) = f_i$ for $i=1,2,3$. We let $(e_1,e_2,e_3)\sim (f_1,f_2,f_3)$ if
$$\mathrm{sgn}(\alpha) := \frac{\det(\alpha)}{|\det(\alpha)|}=1.$$
It is easy to see from the properties of the determinant that $\sim$ is an equivalence relation. Since $\mathrm{sgn}(\alpha) \in \{\pm 1\}$, it is easy to see that there are exactly two equivalence classes, with representatives $[(e_1,e_2,e_3)]$ and $[(e_2,e_1,e_3)]$. The two equivalence classes correspond to the notion of chirality, but what you call each class ("left'' or "right'') is just an arbitrary convention.
In physics one usually uses a "frame", that is, an ordered orthonormal basis, to obtain a system of coordinates, so it is worth noting that this equivalence relation restricts to one on ordered orthonormal bases. Indeed in that case $\alpha$ will be an orthogonal transformation, so that $\det(\alpha) \in \{\pm 1\}$ and hence $\mathrm{sgn}(\alpha)$ is just $\det(\alpha)$. In other words, two frames are in the same equivalence class of orientation if one can go from one to the other by applying an element of $\mathrm{SO}_3(\mathbb R)$. This is equivalent to saying that two frames are equivalent if one can move from one frame to the other through rigid motions.
This last point yields your conjecture, because everyone agrees on what the ordered basis $(e_1,e_2,e_3)$ of $\mathbb R^3$ is. The point of the right-hand rule however, is rather to tell you how you should label the basis relevant to a physical system so that someone using your coordinates will associate a system with the same orientation as the one in the original physical system. In other words, the RHR is not really a device designed to specify an orientation of $\mathbb R^3$, but rather a way of ensuring that people read and assign abstract coordinates of physical systems in a manner which preserves the orientation of that system.
In relation to the question "what is unique about the right hand rule",-- it is just that it provides a rule which specifies to anyone with hands an ordered basis of $\mathbb R^3$ whose orientation is independent of the person using the rule. (Anyone who can use their anatomy to specify a frame in $\mathbb R^3$ can also use the rule of course, provided they compare their frame with the one given by the right-hand rule).
It might also be worth pointing out that the definition of $\sim$ makes sense in $\mathbb R^n$ for any $n$, and that, for any $n$, it has exactly two equivalence classes. If you have decided agreed some convention on what "right-hand" orientation is in $\mathbb R^n$ for all $n$ (e.g. that $(e_1,e_2,\ldots,e_n)$ is "right-handed" for all $n$ say) then if you have a hyperplane $H$ of $\mathbb R^n$,