How to make matrix unitary again?

Question

I know QR decomposition of matrix A, with mxn, where m > n. I want to find a QR decomposition of matrix $$\begin{bmatrix} A \\ a \end{bmatrix}$$ where a - vector 1xn. My idea is: $$\begin{bmatrix} QR \\ aR^{-1}R \end{bmatrix} = \begin{bmatrix} Q \\ aR^{-1}\end{bmatrix}R = \hat{Q}R$$ And it works fine R is stil upper-triangle. But $\hat{Q}$ is not unitary. What should I do?