= Planar projection of an isotropic random vector
{title2=$\mathbb E R_\perp=(\pi/4)\mathbb E R,\quad\mathbb E R_\perp^2=(2/3)\mathbb E R^2$}
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.
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