Rigorous definition of left hand coordinate system.

Question

It seems that for a 3D vector space over $\mathbb{R}$, we cannot define whether a coordinate system if left-handed or right-handed, since we cannot compare an abstract vector with our fingures. So how do we rigorously define "right hand rule" and cross product in an abstract 3D vector space?

Answer 1

You can just say that in $\Bbb R^3$ the "natural" orientation is the right-handed one. The cross product is defined by the identity $\newcommand\va{\mathfrak{a}} \newcommand\vb{\mathfrak{b}} \newcommand\vc{\mathfrak{c}} \newcommand\ve{\mathfrak{e}}$ $$ \det(\va,\vb,\vc)=\langle \va\times \vb,\vc\rangle, ~\text{ or a little informally }~ \va\times \vb=\sum_{k=1}^3\det(\va,\vb,\ve_k)\ve_k= \begin{vmatrix}a_1&b_1&\ve_1\\a_2&b_2&\ve_2\\a_3&b_3&\ve_3\end{vmatrix}. $$ Here the determinant with its anti-symmetric properties, or in other words the epsilon tensor, incorporates the connection to the orientation.

In any visualization of the abstract space the choice of the assignment of the abstract coordinates to the visual basis vectors/coordinate axes determines if the natural orientation of the abstract space is the visually left- or right-handed one.

Answer 2

You can define something like the cross product without choosing an orientation. It's called the wedge product. In a lot of applications, it's the better formalism anyway.

For example, in physics, we use the cross product to talk about torques and magnetic fields. But whether those fields are represented by vectors pointing one way or the other is physically meaningless.

Answer 3

For the reason you state, you can't define the right-hand rule or the cross product for an arbitrary three dimensional real vector space. To do that you need to equip the vector space with some extra structure called an orientation. From Wikipedia:

Let $V$ be a finite-dimensional real vector space and let $b_1$ and $b_2$ be two ordered bases for $V$. It is a standard result in linear algebra that there exists a unique linear transformation $A : V \to V$ that takes $b_1$ to $b_2$. The bases $b_1$ and $b_2$ are said to have the same orientation (or be consistently oriented) if $A$ has positive determinant; otherwise they have opposite orientations. The property of having the same orientation defines an equivalence relation on the set of all ordered bases for $V$. If $V$ is non-zero, there are precisely two equivalence classes determined by this relation. An orientation on $V$ is an assignment of $+1$ to one equivalence class and $−1$ to the other.

Then you say the right-hand bases are the ones in the class assigned $+1$.

(You'll need even more structure if you want the cross product to be uniquely determined. With only an orientation you can say which side of the plane spanned by $v$ and $w$ the vector $v\times w$ will be in, but nothing more.)