Finding orthogonal input patterns associated with logistic function output
All you have to do is complete $\omega$ to an orthogonal $\mathbb{R}^3$ basis, as $\omega^T V = 0$ and $v_1$, $v_2$ are orthogonal. You can do this using Gram Schmidt orthogonalization. The two other vectors of the basis will be $v_1$ and $v_2$.

Edit:

The initial vectors to do the Gram Schmidt process can be any pair of independent vectors, for instance $(1, 0, 0)$ and $(0, 1, 0)$.