As pointed out in the comments, the matrix has nothing to do with the question, and you are simply trying to compute the distribution of $x \cdot v$ as $x$ varies over the unit sphere, and $v$ is a fixed unit vector. By rotational invariance, you might as well assume that $v = (1, 0, \dots, 0),$ at which point the question becomes an easy integration exercise.

As pointed out in the comments, the matrix has nothing to do with the question, and you are simply trying to compute the distribution of $x \cdot v$ as $x$ varies over the unit sphere, and $v$ is a fixed unit vector. By rotational invariance, you might as well assume that $v = (1, 0, \dots, 0),$ at which point the question becomes an easy integration exercise.

Edit if you don't want to bother integrating, the concentration of measure phenomenon (first observed by Boltzmann, I believe) is that the distribution of the areas of cross sections of the sphere is essentially normal for moderate $n.$ ( you can easily compute the variance), so then you can approximate the answer to your question by an inverse error function.