= Solution
The <reciprocal lattice> is the set of <wavevectors> $\mathbf G$ satisfying $e^{i\mathbf G\cdot\mathbf R}=1$ for every direct <Bravais lattice> translation $\mathbf R$. Equivalently, $\mathbf G\cdot\mathbf R$ is an integer multiple of $2\pi$. It describes the spatial periodicities that can produce coherent <Bragg scattering>.
For direct primitive vectors $\mathbf a_i$ and oriented cell volume $\Omega=\mathbf a_1\cdot(\mathbf a_2\times\mathbf a_3)$, the primitive reciprocal vectors are
$$
\boxed{\mathbf b_1=2\pi\frac{\mathbf a_2\times\mathbf a_3}{\Omega},\qquad
\mathbf b_2=2\pi\frac{\mathbf a_3\times\mathbf a_1}{\Omega},\qquad
\mathbf b_3=2\pi\frac{\mathbf a_1\times\mathbf a_2}{\Omega}.}
$$
They obey $\mathbf a_i\cdot\mathbf b_j=2\pi\delta_{ij}$, and every reciprocal vector is an integer combination of them. Their units are inverse length. The reciprocal primitive cell volume is $(2\pi)^3/|\Omega|$, so expanding a direct lattice contracts its reciprocal scale.
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