Angle between two random unit vectors uniformly distributed

Question

Consider $x, y \in S_{1}^{d-1}$ (the unit n-sphere in d dimensions) with $(x \cdot y)^2 = 1/d$. I need to compute the angle $\alpha$ between $x$ and $y$ for $d$ = $3$ and asymptotically for large $d$. I have prove that $E[(x \cdot y)^2] =1/d$ but I don´t know how can I compute the angle.

Answer 1

If $d$ is large enough, $\mathbb{E}[(x\cdot y)^2]\approx 0$, therefore $\cos^2\alpha\approx0$. By using Taylor expansion at $\alpha=\pi/2$, which is

$$\cos^2\alpha=(\alpha-\pi/2)^2-1/3(\alpha-\pi/2)^4+\cdots$$

We get

$$\mathbb{E}[\cos^2\alpha]\approx\mathbb{E}[(\alpha-\frac{\pi}{2})^2]\approx 1/d$$