I have a question that might be answered with a pointer to some references or with some discussion. I did some searching, to no avail, but I realized that I might not have the vocabulary to form a successful search.

Suppose we have a random (or pseudorandom) vector $r\in\mathbb{R}^{n}$ with $\left\|r\right\|=1$, e.g. generated by the rand() function in Matlab and normalized. Let $S\in\mathbb{R}^{n\times n}$ be a symmetric matrix with the orthogonal decomposition $S=V^{T}DV$ where the columns of $V$ are orthonormal and $D$ is the diagonal matrix of eigenvalues. Let $v_{i}$ be a column of $V$. What is the probability that $\left|r^{T}v_{i}\right|< \alpha$ where $\alpha\in\left(0,1\right)$. In other words, what is the probability that $r$ has only a certain size component in a particular eigenvector?