Test of handedness

Question

I'm reading a book on linear algebra, where the author gives a method to test the handedness or chirality of a given set of 3 basis vectors.

if $(v1 \times v2) \cdot v3 > 0$ then it's right-handed, while if it's less than $0$, it's left handed.

What beats me is that numbers are just numbers, left or right handedness of a system depends on the viewer and how he interprets the given data.

Taking the canonical basis vectors $\hat i, \hat j, \hat k$, in both left and right handed systems $i \times j = k$, thereby $k \cdot k = \lVert k\rVert ^2 > 0$ (always), then how does this test hold true?

Answer 1

Orientation (handedness) is not about a set of vectors, it is about an ordered list of vectors. That is, a certain ordering, $(i,j,k)$ is agreed to as right handed. Then $(j,i,k)$ is left handed. This may or may not agree with some notion you have from physics, hard to predict.

A smooth manifold is orientable...never mind.

Answer 2

I think there is an assumption behind your quoted statement, that is, the basis vectors used to express $v_1$, $v_2$ and $v_3$ are in a right-handed coordinate system. Then everything makes more sense. So when you define arbitrarily three vectors (e.g. $v_1$, $v_2$ and $v_3$), $(v_1\times v_2)\cdot v_3$ will tell if the three vectors have the "same" handedness or the "opposite" handedness as compared to the handedness of the basis vectors.

For example, if you define $v_1=[1,0,0]$, $v_2=[0,1,0]$ and $v_3=[0,0,1]$, which are basically equivalent to the basis vectors, the handedness test $(v_1\times v_2)\cdot v_3$ is positive as expected since your basis vectors are by assumption in a right-handed coordinate system.

And when your twist the above example a bit by defining $v_1=[1,0,0]$, $v_2=[0,1,0]$ and $v_3=[0,0,-1]$, the handedness test $(v_1\times v_2)\cdot v_3$ is negative.

So your quoted statement is better written as:

Given that v1, v2 and v3 are defined in a right-handed coordinate system, if (v1×v2)⋅v3>0 then it's right-handed, while if it's less than 0, it's left handed.