There are two velocities on the sphere. How to sum them on the sphere? As shown in the figure, there are two velocities on the sphere, one is the velocity along the meridian direction, the arc length is its size, the other is the velocity along the equatorial direction, and the arc length is its size. How to sum on a sphere? Can we use the parallelogram rule on the sphere?
What I'm talking about is the velocity on a two-dimensional sphere.

 A: You need to learn about the concept of tangent vectors and the tangent plane. If these two vectors are located at the same point, then we think of them as living in the plane tangent to the sphere at this point, and you add them as you would with any vectors in a plane.
If they're at different points on the sphere, then they live in different tangent spaces, and they can't be added. In order to add them, you would need to prepare them by moving one to the same place as the other, but this kind of parallel transport is path-dependent, so there's no unambiguous way to define how to do this.
A: For your problem, you have to calculate the geodetic equations for a sphere.
The sphere position vector with sphere radius equal one is: 
$$\vec{R}=  \left[ \begin {array}{c} \sin \left( \theta \right) \cos \left( 
\varphi  \right) \\ \sin \left( \theta \right) \sin
 \left( \varphi  \right) \\ \cos \left( \theta
 \right) \end {array} \right] 
\tag 1$$
from equation (1) you get the geodetic equations for $\theta(s)$ and $\varphi(s)$ where $s$ affine path parameter
$${\frac {d^{2}}{d{s}^{2}}}\theta \left( s \right) -\cos \left( \theta
 \left( s \right)  \right) \sin \left( \theta \left( s \right) 
 \right)  \left( {\frac {d}{ds}}\varphi  \left( s \right)  \right) ^{2
}=0
\tag 2$$
and 
$${\frac {d^{2}}{d{s}^{2}}}\varphi  \left( s \right) +2\,{\frac {\cos
 \left( \theta \left( s \right)  \right)  \left( {\frac {d}{ds}}
\varphi  \left( s \right)  \right) {\frac {d}{ds}}\theta \left( s
 \right) }{\sin \left( \theta \left( s \right)  \right) }}=0
\tag 3$$
for a given initial condition,you can solve numerically  equation (2) and (3)  and get $\theta(s)\,,\varphi(s)$ and with equation (1) the path on the sphere.

the red line is the solution for the initial condition  velocity $D(\theta)(0)=V/R\,,D(\varphi)(0)=0$,   the blue line is the solution for the initial condition  velocity $D(\theta)(0)=0\,,D(\varphi)(0)=U/R$.
the solution for both the initial conditions $D(\theta)(0)=V/R$ and $D(\varphi)(0)=U/R$ is the green line, which is a great circle on the sphere
