Question

What is the dimensions of solid angle and unit vector

Answer 1

Answer:

The solid angle is defined as the area on the unit sphere subtended by the angle divided by one unit area. It's a ratio so it's a single dimensionless number.

I see why you think it should be a 2D quantity, because the surface of a sphere, and any patch on it, is a 2D manifold and you need two quantities (traditionally θθ and ϕϕ) to map it. When you calculate an area on the sphere you are basically calculating a definite integral over θθ and ϕϕ, and the result is of course just a single number. You do lose information in the process - for example you just know the total area not the shape of the patch on the sphere.

The solid angle that covers the whole sphere is of course 4π4π/1 or 4π4π.