2. Planes and the reciprocal lattice. In the lectures I claimed that the reciprocal lattice vector
$G_{n_1n_2n_3} = n_1b_1 + n_2b_2 + n_3b_3$
is perpendicular to the plane ($n_1n_2n_3$). This is always true, and does not require the assumption that the
primitive lattice vectors are orthogonal. Here you will prove this claim.
(a) First prove this statement for the case where two of the integers ($n_1n_2n_3$) are vanishing, e.g. $n_2 =$
$n_3 = 0$.
Hint: the plane is spanned by the vectors $a_2$ and $a_3$. Find a vector perpendicular to $a_2$ and $a_3$ (i.e.
normal to the plane), and show that it is parallel to the reciprocal lattice vector $b_1$.
(b) Repeat this calculation for the case where only one of the integers is vanishing, e.g. $n_3 = 0$.
Hint: The vector $a_3$ is clearly parallel to the plane. To find a second vector, consider the intersections
of the plane with the axes defined by $a_1$ and $a_2$.
(c) Finally, using the approach in part (b), prove this statement for the case where all the integers are
nonzero.
