Let $\mathbb{S}^n=\{\mathbf{x}\in\mathbb{R}^{n+1}\colon\lVert\mathbf{x}\rVert^2=1\}$ be the unit n-Sphere and $\mathbf{x}\in\mathbb{S}^n$.
Given a vector $\mathbf{a}\in\mathbb{R}^{n+1}$ and an angle $\theta\in[-\pi,\pi)$, we need to find a point $\mathbf{y}\in\mathbb{S}^n$ such that it belongs to both the n-Sphere surface and on the geodesic $g_a$, as shown in the figure below.
Edit: I will update the question asap with an answer I came up with after getting help from another question.