Spherical line through two points

Question

Given two vectors $u$ and $v$ in unit sphere $S^2$, I need to find the point $w\in S^2$ such that $L_w$ is the line through $u$ and $v$. My question is how can there be the point $w$ if a spherical line through $u$ and $v$ is supposed to contain infinitely many points $w$. Or am I not on the same page with something?