Given two orthogonal unit vectors a $ \in \mathbb{R}^N$ and b $ \in \mathbb{R}^N$ such that a$^T$ b $= 0$, |a|$=1$, |b|$=1$ where a$^T$ means the transpose of a, and a vector w$\in\mathbb{R}^{1\times N}$. Suppose that each element in w follows the Gaussian distribution $\mathcal{N}(0,1/N)$. Define a random variable as $z =$ (wa)(wb), and what kind of distribution does $z$ follow?
I have run the Monte Carlo simulations, and it seems that $z$ follows $\mathcal{N}(0,1/N^2)$. How to evaluate such a distribution in a theoertically manner?