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&gt;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 &gt; 0 (always), then how does this test hold true?

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I&#039;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&amp;gt;0 then it&#039;s right-handed, while if it&#039;s less than 0, it&#039;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    &amp;gt;  0 (always), then how does this test hold true?

Jordin Church · Accepted Answer

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.

I'm reading a book on linear algebra, where the author gives a method to test the handedness or chir

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