Let $s = (\frac{1}{\sqrt{d}}, \ldots, \frac{1}{\sqrt{d}})$ and $u \in \mathbb{R}^d$ be two distinct unit norm vectors in the first orthant. Consider moving along the great circle defined by $s$ and $u$ (in the direction from $s$ to $u$) until the first intersection of the great circle and one of the following hyperplanes $\{x_1 = 0, x_2 = 0, \ldots, x_d = 0\}$ is reached. Let $v$ denote that intersection.
Is it possible to explicitly give an equation for the point $v$ or at least the angle between $u$ and $v$?