Let $Z\sim N(0,I_{n})$ be a standard normal Gaussian random vector in $\mathbb{R}^{n}$, and let $V_1$ and $V_2$ be two isotropic and independent random vectors in $\mathbb{R}^n$. They also have the same distribution, although I'm not sure if this matters. Can we say anything about the projection values $\left<Z, V_1\right>$ and $\left<Z, V_2\right>$?
I know that if $V_1$ and $V_2$ are fixed orthogonal vectors, then the projection values are independent. What if they're not? Thank you in advance!