= Solution
For <elastic scattering>, $|\mathbf k_{\rm in}|=|\mathbf k_{\rm out}|=2\pi/\lambda$. Contributions from sites differing by any <Bravais lattice> translation $\mathbf R$ add in phase exactly when
$$
e^{i(\mathbf k_{\rm out}-\mathbf k_{\rm in})\cdot\mathbf R}=1
$$
for every $\mathbf R$, namely when the momentum-transfer wavevector is a <reciprocal lattice> vector $\mathbf G$. For a family of planes of spacing $d$, the $n$th reciprocal vector normal to those planes has magnitude $2\pi n/d$. The scattering geometry gives $|\mathbf G|=2k\sin\theta$. Combining the two expressions proves
$$
\boxed{2d\sin\theta=n\lambda.}
$$
Conversely, reflection from that plane family satisfying <Bragg's law> supplies the corresponding normal reciprocal vector. The scalar angle condition alone needs this plane-normal geometry; it does not specify an arbitrary outgoing direction. A nontrivial atomic basis can extinguish an otherwise allowed reciprocal-lattice reflection through its <structure factor>.
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