All the matrices that are orthogonal and have $q_1,q_2$

Question

"Determine all the orthogonal matrices $Q=[q_1,q_2,q_3]$ that have as the first two columns the vectors $q_1=\frac{1}{\sqrt{6}}(-1,2,-1)^T, \ q_2=\frac{1}{\sqrt{3}}(1,1,1)^T$".

I used the cross-product $q_1 \times q_2=\begin{bmatrix} \frac{3}{\sqrt{18}} \\ 0 \\ \frac{-3}{\sqrt{18}} \end{bmatrix}$

This is an orthogonal vector. Put in the matrix $Q$, $Q$ is orthogonal. Are there any other vectors that could be found and still keep $Q$ orthogonal?

Answer 1

Assuming that only real values are allowed, $\pm q_1\times q_2=\mp q_2\times q_1$ are the only solutions.

Answer 2

All possible vectors are $\pm q_1\times q_2$, one for each connected component of $O(3)$.