The derivation of reciprocal lattice vectors in terms of the direct space lattice vectors starts by applying expanding a translationally invariant lattice function $f(\bf{R_k}+r)$ in plane waves $f_k e^{i G_m \cdot R_k} e^{i G_m \cdot r} $. Then by the translational invariance
$$ e^{i G_m \cdot R_k} = 1 $$ from which we have (1) $$ G_m \cdot R_k = 2\pi N $$ where N is an integer.
From this the next step in most derivations says that (2)
$$ \vec{a_i}^* \cdot \vec{a_j} = 2\pi \delta_{i,j} $$ or in matrix form $$ (\bf{A^*})^T\bf{A} = 2\pi \bf{I}\\ (\bf{A^*})^T = 2\pi \bf{A}^{-1}. $$ However, I don't see how we can deduce (2) from (1).
Writing $G_m = h \vec{a_1}^* + k \vec{a_2}^* + l \vec{a_3}^*$ and $R_k = m \vec{a_1} + n \vec{a_2} + o \vec{a_3}$ for $$ (h \vec{a_1}^* + k \vec{a_2}^* + l \vec{a_3}^*) \cdot (m\vec{a_1} + n \vec{a_2} + o \vec{a_3}) = 2\pi N $$ I still don't see it immediately.
Any help would be appreciated, thank you.