Test of handedness 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?

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.