Finding orthogonal input patterns associated with logistic function output

Question

I've been given this problem but cannot seem to get an analytical solution. I've tried satisfying the logistic function with several vectors but have difficulty finding ones which are also orthogonal. The problem begins with me being given a neural unit weight vector, $$w =\begin{bmatrix} 1 \\ \frac{1}{4} \\ \frac{1}{9} \\ \end{bmatrix}$$ The neuron output u is related to its input pattern v∈R3 by $$u = f(w^Tv + 1)$$ where $$f(x) = (1 + e^{-x})^{-1}$$ To find a pair of orthogonal vectors, v1 and v2, for which the unit output $u_{i} = 0.5, i=1,2.$

So far I know that $w^TV = 0$ for the function output to be 0.5.

Answer 1

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 $(0, 1, 0)$.