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 i ^ , j ^ , k ^ in both left and right handed systems i × j = k, thereby k k = k 2 > 0 (always), then how does this test hold true?
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Answers (1)

Jordin Church
Answered 2022-07-13 Author has 11 answers
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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