First I'd like to give you the context of my question (you can skip this paragraph if you like).
In this paper on SIC-POVM, the authors show sets of fiducial vectors (section IV.) used to explicitly construct SIC-POVM in dimensions $d=2,3,4$. I would like to know, whether it is possible to find orthogonal vectors in a given fiducial set. For $d=2$ the fiducial set itself consists of two orthogonal vectors. But for $d=3$ it is much less obvious.
Now coming to my question:
For $r_0$ satisfying $1/\sqrt{2}<r_0\leq \sqrt{2/3}$, define
$$
r_\pm(r_0)=\frac{1}{2}r_0\pm\frac{1}{2}\sqrt{2-3r_0^2}.
$$
Then define the set of vectors
$$
V=\left\{
\begin{pmatrix}
r_0\\ r_+ e^{i\theta_1}\\ r_- e^{i\theta_2}
\end{pmatrix},
\begin{pmatrix}
\text{plus all vectors formed}\\
\text{by permuting of elements}
\end{pmatrix}
\middle|
\theta_1,\theta_2\in \left\{ \frac{\pi}{3}, \pi,\frac{5\pi}{3}\right\},\frac{1}{\sqrt{2}}<r_0\leq \sqrt{\frac{2}{3}}
\right\}
\cup
\left\{
\begin{pmatrix}
1/\sqrt{2}\\ e^{i\theta_1}/\sqrt{2}\\ 0
\end{pmatrix},
\begin{pmatrix}
\text{plus all vectors formed}\\
\text{by permuting of elements}
\end{pmatrix}
\middle|
0\leq \theta_1 < 2\pi
\right\}.
$$
If possible, find a set $S=\{v_1,v_2,v_3\}$ of vectors $v_i\in V\subset\mathbb{C}^3$, that are mutually orthogonal, i.e. $v_1^*\cdot v_2=v_2^*\cdot v_3=v_1^*\cdot v_3=0$.
As you can see this set is fairly complicated and it is not obvious that the set $S$ exists within $V$. I have tried out Gram-Schmidt starting with two orthogonal vectors from $V$ to find the third orthogonal vector, but the result was always outside of $V$. Of course I could write down an explicit system of equations and try to solve for all the variables. But this seems very tedious because of all the possible permutations of components. Furthermore the resulting equations are not linear.
Since I don't have experience with these kind of problems I am wondering if there are any kinds of tricks, that avoid solving a complicated system of equations. If you find $S$ just by looking at $V$ it would also help.
Cheers,
ncsdn