Assume that we have a sphere in 3D space. Let point $A$ rest on the surface of the sphere. I can apply a force $F$ of any magnitude and any direction, so long as it points tangential to the surface and originates from point $A$. Then, at various points along the sphere, forces of different magnitudes are applied tangential to the surface. I have created a diagram, shown below:
The goal is to find a magnitude and direction of the force such that there is no net torque on the ball.
My question is this: Can I simply decompose the forces into its $x$, $y$, and $z$ components, look at the ball along each axis, treat it as a 2D situation and solve for the force and direction of $F$? This is the only way that I can think about doing it, as the only prior experience with torque I have is in 2D. This doesn't quite make sense to me as rotating an object along the $x$ axis and then along $y$ is different than rotating it around $y$ and then $x$. Additionally, to me at least it makes sense to only have one axis of rotation, even in three dimensions...
If not, then how would I go about solving this? I am comfortable with linear algebra (vector addition, dot products, cross products, ect...) and understand that I might have to use them in the solution to this problem.
Edit: If this helps, I know that this case is pretty trivial to solve when the forces all point towards $A$, along the sphere's surface. I just cancel out the magnitudes of each of the other forces with $F$ much like a translational motion problem. Am I on the right track to a solution or no?