How to define a vector along a specific direction?

There is a total of 16416 points which can be defined by (x, y, z) in cartesian coordinates or (theta, phi, r) in spherical coordinates. All of these points are located on the surface of a sphere. For each point, I want to define a vector go through it. So there is a total of 16416 vectors. I want to simulate the following three cases:

Case-1: Circumferential. For example, all vectors are tangent to the latitude of the sphere (Figure, left).

Case-2: Helicoidal. For example, all vectors are tangent to the helical line of the sphere (Figure, middle).

Case-3: Meridional. For example, all vectors are tangent to the meridian of the sphere (Figure, right).

Note: When the tangent direction was determined, it does not matter the vector points towards which end. Just make sure that the directions of all vectors are consistent.

My question is: How to define a single parameter (i.e., angle) so that I can change the direction of all vectors by adjusting this parameter? For example, I can simulate three different cases by assigning three different parameters. Thank you.