= Solution
{c}
In <elastic scattering>, the incoming and outgoing <wave vectors> have equal magnitude $k_0=2\pi/\lambda$. Their mutual angle is $2\theta_{hkl}$. The <scattering vector> $\mathbf q=\mathbf k_f-\mathbf k_i$ therefore has magnitude
$$
q^2=2k_0^2\left[1-\cos(2\theta_{hkl})\right]
=4k_0^2\sin^2\theta_{hkl}.
$$
The <elastic Bragg scattering condition> is $\mathbf q=\mathbf G_{hkl}$, giving
$$
\boxed{G_{hkl}=\frac{4\pi}{\lambda}\sin\theta_{hkl}.}
$$
Equivalently, $G_{hkl}=2\pi/d_{hkl}$ turns this relation into <Bragg's law>, $2d_{hkl}\sin\theta_{hkl}=\lambda$, with the reflection order incorporated into the <Miller indices>.
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