Lattice vectors calculation
You can find this using elementary geometry. Also, since the unit cell may be positioned in different ways with respect to the coordinate system, there can be many different answers, all equally valid.

One possible answer implies alignment of the cell vectors $\vec a$ and $\vec c$ with the coordinate axes X and Z, which leads to the following:
$$\left\{\begin{align}
\vec{\bf a} =&amp; \Big(|a|,0,0\Big)\\
\vec{\bf b} =&amp; \left(-{1\over2}|a|,{\sqrt3\over2}|a|,0\right)\\
\vec{\bf c} =&amp; \Big(0,0,|c|\Big)
\end{align}\right.
$$