Finding a vector field perpendicular to the surface of a sphere

[walking_edges]

I'm trying to figure out if a given vector field is perpendicular at the surface of a sphere of radius R. The vector field is given in spherical coordinates.

I initially attempted to take the cross product of the vector field with the normal vector at the surface of the sphere to see if it was zero, but unfortunately, the cross product in spherical coordinates is much too difficult to work with. I'm hoping to find an alternative method.

Does anyone know how to find out if a vector field is perpendicular to a surface without using the cross product?

 

[LCKurtz]

At each point on the sphere the field would have to be a multiple of the radius vector. Something like

$$\vec F = f(\theta,\phi)\vec \rho$$

where f is a scalar.

 

[walking_edges]

Thank you very much!