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 $\hat{i},\hat{j},\hat{k}$ in both left and right handed systems $i\times j=k$, thereby $k\cdot k=\Vert k{\Vert}^{2}>0$ (always), then how does this test hold true?

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 $\hat{i},\hat{j},\hat{k}$ in both left and right handed systems $i\times j=k$, thereby $k\cdot k=\Vert k{\Vert}^{2}>0$ (always), then how does this test hold true?