 # I'm reading a book on linear algebra, where the author gives a method to test the handedness or chir Joshua Foley 2022-07-12 Answered
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×v2)⋅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 $\stackrel{^}{i},\stackrel{^}{j},\stackrel{^}{k}$ in both left and right handed systems $i×j=k$, thereby $k\cdot k=‖k{‖}^{2}>0$ (always), then how does this test hold true?
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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.

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