(Solved): The unit cell for a primitive hexagonal lattice can be described based on Cartesian coordination sy ...

The unit cell for a primitive hexagonal lattice can be described as the following based on Cartesian coordination system: $\begin{array}{l}\vec{a}=[100]=a?\vec{x}\\ \vec{b}=[010]=\cos 120^{?}?a?\vec{x}+\sin 120^{?}?a?\vec{y}\\ \vec{c}=[001]=c?\vec{z}\end{array}$ For its reciprocal lattice, what are the basis vectors ${\vec{a}}^{?},{\vec{b}}^{?}$, and ${\vec{c}}^{?}$ of this unit cell? Describe them using the variables $a$ and $c$ based on $\vec{x}?\vec{y}?\vec{z}$ axes. Which one of the 14 Bravais lattices it is?