If $U$ is uniformly distributed on $S^{d-1} \subset \mathbb{R}^d$, what's the distribution of its orthogonal projection onto any vector? Let $U \in S^{d-1} \subset \mathbb{R}^d$ follow a uniform distribution on a sphere. Let $v \in  \mathbb{R}^d.$ Then is the orthogonal projection $U^{T}v=\langle U,v \rangle$ uniformly distributed, and if yes/no, how do I go about proving it? This is motivated by this relevant question for uniform distributions on 2D spheres. If it's wrong, could you give a counterexample?
If it's not uniformly distributed, how can we find the PDF of $U^{T}v=\langle U,v \rangle?$
 A: Let $v$ be a unit vector.
Since the orthogonal group acts transitively on the unit sphere, there exists a rotation matrix $R$ such that $Rv=e$, where $e=(1,0,\ldots,0)'$.
Now let $Z=(Z_1,\ldots,Z_d)'$ be a vector of independent identically distributed $N(0,1)$ random variables.
The distribution of $Z$ is clearly rotationally invariant, and the normalized vector $U=Z/\|Z\|$ is uniformly distributed over the unit sphere, as is $R'U$. (See, eg, the answer by Henricus V. to this MSE question, and comments, and this one, about this trick. It is a converse to Maxwell's theorem.)
The quantity $\langle U, v\rangle=\langle U,Re\rangle=\langle R' U,e\rangle,$ so the distribution of  $\langle U, v\rangle$
is the same as that of $\langle U, e\rangle$.
So we may as well assume $v=e$ in working out the distribution of $T=\langle U,v\rangle$.
With this notation,  $T=\langle U,v\rangle=Z_1/\sqrt{\sum_{i=1}^d Z_i^2}$, and $T^2=Z_1^2/\sum_{i=1}^d T_i^2$.
The quantities $Z_i^2$ are iid Gamma distributed: $Z_i^2\sim\Gamma(\frac 1 2,\frac 1 2)$ and $\sum_{i=2}^d Z_i^2\sim\Gamma(\frac{d-1}2,\frac 12)$ and hence $T^2$ has the $\operatorname{Beta}(\frac 1 2,\frac {d-1}2)$  distribution.
If we write $W=T^2$, the density function of $W$ is proportional to $w^{-\frac{1}2} (1-w)^{\frac{d-1}2-1}$ for $0<w<1$, and the density of $T$ is  proportional to $(1-t^2)^{(d-3)/2}$ for $-1<t<1$. Only if $d=3$ does the distribution of $T$ become uniform over its range.
