An Inner Product in Non-Euclidean Space

Classical Mechanics

Two vectors $a^i = (1,2,3)$ and $b^i = (2,3,4)$ live in a space with spatial metric given by the invariant interval

$ds^2 = 2dx^2 + \frac12 dy^2 + dz^2.$

Compute the inner product $a_i b^i = g_{ij} a^i b^j$ .