Completing a Rotation Matrix

A $3 \times 3$ matrix has its first two rows fixed and its third row unknown:

$Q = \frac{1}{3}\begin{pmatrix} 2 & -2 & 1 \\ 1 & 2 & 2 \\ a & b & c \end{pmatrix}$

The scalars $a, b, c$ are chosen so that $Q$ is an orthogonal matrix ($Q^{\mathsf{T}}Q = I$) and $\det Q = +1$, i.e. $Q$ is a pure rotation rather than a reflection.

What is $(a,\ b,\ c)$?