Planar projection of an isotropic random vector ($\mathbb{E}{R}_{\perp }=(\pi /4)\mathbb{E}R,\mathbb{E}{R}_{\perp }^2=(2/3)\mathbb{E}{R}^2$)

For an isotropic three-dimensional random vector, its direction is uniform on the sphere and independent of its length. Projection onto a fixed plane gives $R_{\perp }=R\sin \theta$. The solid-angle averages are $\mathbb{E}\sin \theta =(1/2){\int }_0^{\pi }{\sin }^2\theta d\theta =\pi /4$ and $\mathbb{E}{\sin }^2\theta =(1/2){\int }_0^{\pi }{\sin }^3\theta d\theta =2/3$. This proves the two formulas whenever the corresponding moments exist. The mean-length factor and root-mean-square factor are different. Both are independent of the radial distribution, so projection preserves a polymer's chain-length scaling exponent.