1. The problem statement, all variables and given/known data Consider a 2-dimensional line element: [tex]ds^2=x^2dx+2dxdy-dy^2[/tex] Raise and lower the indices of given vectors by finding the raised index metric, etc, blah blah 2. Relevant equations [tex]V_a = (1,-1)[/tex] and [tex]W^a = (0,1)[/tex] 3. The attempt at a solution Solution is given and I understand how to raise and lower indices, but I'm just wondering if my book has a massive exemplary typo. They raise V by multiplying it by the raised metric, which is fine. I get that. I don't get why they use the raised metric for lowering the W vector. Shouldn't you just use the lower metric given by the line element?