<s>[INST] Find a 3 x 3 orthogonal matrix Find a 3x3 orthogonal matrix whose first column is $[\frac1{\sqrt{3}},{- \frac 1{\sqrt{3}}}, \frac1{\sqrt{3}}]$.
I was thinking this has to do with: ${Q}^T•Q=I_{n}$, but I still don't understand how to go about this.
 [/INST] In this case we can also guess $3$ vectors without calling for Gram-Schmidt by noticing
$\begin{pmatrix} 1 \\ -1 \\ 1\end{pmatrix}\text{ , }\begin{pmatrix} 1 \\ 2 \\ 1\end{pmatrix}\text{ and }\begin{pmatrix} 1 \\ 0 \\ -1\end{pmatrix}\text{ are orthogonal.}$
By keeping first and third coordinate of $v_1,v_2$ equal, we were able to find an easy $v_3$.
Normalizing this gives $Q=\frac 1{\sqrt{6}}\begin{pmatrix} \sqrt{2} & 1 & \sqrt{3} \\ -\sqrt{2} & 2 & 0 \\ \sqrt{2} & 1 & -\sqrt{3} \end{pmatrix}$
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