Orthogonality & Least Squares · Hard problem

A $3\times3$ orthogonal calibration matrix $Q$ (rows orthonormal, $\det Q = 1$) has first two rows $\mathbf r_1 = \tfrac13(2,2,1)$ and $\mathbf r_2 = \tfrac13(2,-1,-2)$. Using $\mathbf r_3 = \mathbf r_1\times\mathbf r_2$ to complete the basis and $Q^{-1}=Q^T$ to solve $Q\mathbf x = (1,2,2)$, find the third coordinate $x_3$ of $\mathbf x$.