Reciprocal basis and orientation

[feynman137]

How to prove that two reciprocal basis are either both right ended or both left-handed? If (e_1,e_2,e_3) and (e^1,e^2,e^3) are two such basis, since the scalar triple products depend on orientation, it would be enough to show that VV'=1 (where V and V' are the volumes, taken with their sign, of the two parallelepipeds obtained from the two basis vectors). How to do it?

 

[feynman137]

I figured it out: just write (e_2 x e_3).(e^2 x e^3) as ((e_2 x e_3) x e^2).e^3 and by basic identities we can prove that VV'=1>0, so the two basis have the same orientation.