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 \(a_i b^i = g_{ij} a^i b^j\).

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