I have a sphere with its center at $(0,0,0)$ and 2 vectors $\vec a$ and $\vec b$ and both of them have a length of spheres radius
$\Vert \vec a \Vert = \Vert \vec b \Vert =R$
Angle between these vectors ($\alpha$) is also known.
If $\vec a$ circles around the $\vec b$ keeping the angle $\alpha$ unchanged we are kind of getting a cone inside the sphere like in this picture.
How can I calculate the coordinates of $\vec a$ if it would rotate $\beta$ degrees around the $\vec b$ ?