I need to fill in the missing entries of $Q$ to make $Q$ an orthogonal matrix. I have no idea how to solve this problem out. I was hoping for some hints on how to go about this.
Problem: $$Q= \begin{bmatrix} 1/\sqrt2 & 1/\sqrt3 & * \\ 0 & 1/\sqrt3 & * \\ -1/\sqrt2 & 1/\sqrt3 & * \\ \end{bmatrix} $$
I know that, just by looking at the problem that $$V_1= \begin{bmatrix} 1 \\ 0 \\ -1\\ \end{bmatrix} $$ and $$V_2= \begin{bmatrix} 1 \\ 1 \\ 1\\ \end{bmatrix} $$
To get a third vector that is orthogonal to both $\vec v_1$ and $\vec v_2$, we take their cross product: $$ \vec v_1 \times \vec v_2 = \begin{vmatrix} \vec i & \vec j & \vec k \\ 1 & 0 & -1 \\ 1 & 1 & 1 \\ \end{vmatrix} = \vec i - 2\vec j + \vec k = \begin{bmatrix} 1 \\ -2 \\ 1 \end{bmatrix} $$ Normalizing, we obtain: $$ \begin{bmatrix} \tfrac{1}{\sqrt 6} \\ \tfrac{-2}{\sqrt 6} \\ \tfrac{1}{\sqrt 6} \end{bmatrix} $$