Here's the question you clicked on:
goodz
Is this a subspace of R3? The set of all vectors of the form <a,b,c> where a+b+c = 0
I think you misunderstood the question. a, b, and c are components of the vector, not vectors themselves. This set of vectors forms a plane through the origin, and therefore constitutes a subspace.
You can easily check the requirements for a subspace, also -- i.e. multiply any such vector by a scalar and its components will still add to zero, and the sum of any two such vectors is another vector whose components add to zero.