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

"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?

• What's the definition of an orthogonal matrix? Since $q_3$ has to be orthogonal to $q_1$ and $q_2$, what does that tell you? (Since $q_1$ and $q_2$ span a plane, what geometric figure is orthogonal to a plane in $\mathbb{R}^3$?) The other requirement for an orthogonal matrix is that the columns have unit length. Combined with what you know from the previous questions, how many possibilities are there for $q_3$? You found one possibility. ($q_3 = q_1 \times q_2$ does work.) How do you find the rest? – Michael Joyce Jan 19 '12 at 18:56

## 2 Answers

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

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

• Sorry Andreas, I haven't seen your answer. – emiliocba Jan 19 '12 at 18:53
• no problem, +1/2 for each connected component – draks ... Jan 19 '12 at 18:57
• but $O(3)$ has two connected component – emiliocba Jan 19 '12 at 23:35
• $2\times \frac{1}{2}=1$ – draks ... Jan 20 '12 at 7:36