I don't really understand what the dimension of a vector space for planes is. Is it 2 or 3 and why? What's the difference between the dimension of the plane as a surface (dimension of all surfaces is 2) and the dimension of plane in a vector space? Also, Wikipedia says that only planes passing through the origin is a subspace of R-3, so does this mean that such plane is equal to R-2? What of planes that do not satisfy this?