An Inner Product in Non-Euclidean Space

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

ds2=2dx2+12dy2+dz2.ds^2 = 2dx^2 + \frac12 dy^2 + dz^2.

Compute the inner product aibi=gijaibja_i b^i = g_{ij} a^i b^j.

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