The primitive translation vectors of the hexagonal space lattice may be taken as a1 = √3 * a / 2 x̂

The primitive translation vectors of the hexagonal space lattice may be taken as
a1 = √3 * a / 2 x̂ + a/2 ŷ ;
a2 = - √3 * a / 2 x̂ + a/2 ŷ ;
a3 = cẑ,
where x̂, ŷ, and ẑ are unit vector in the Cartesian coordinate, and a and c are lattice constants.
(a) Show that the volume of the primitive cell is √3 / 2 * a²c.
(b) Show that the primitive translations of the reciprocal lattice are
b1 = 2𝝅 / √3 * a x̂ + 2𝝅 / a ŷ,
b2 = - 2𝝅 / √3 * a x̂ + 2𝝅 / a ŷ,
b3 = 2𝝅 / c ẑ,
so that the lattice is its own reciprocal, but with a rotation of axes.

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