# An Inner Product in Non-Euclidean Space

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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