Lattice vectors calculation How can I find the crystal lattice vectors, given the crystal parameters?
For example, consider a crystal of rhombohedral symmetry, where a=b and c are known, and alpha=beta=90, gamma=120. How can the lattice vectors be found?
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:
$$\left\{\begin{align}
\vec{\bf a} =&amp; \Big(|a|,0,0\Big)\\
\vec{\bf b} =&amp; \left(-{1\over2}|a|,{\sqrt3\over2}|a|,0\right)\\
\vec{\bf c} =&amp; \Big(0,0,|c|\Big)
\end{align}\right.
$$