I need to determine whether all vectors (a,b,c) for which a+b+c=0 are a subspace of R^3.

I believe it is under scalar multiplication:

k(a,b,c)= (ka, kb, kc)

(ka + kb +kc) = 0

let a= 1, b=2, c=-3, k=5

(5(1) + 5(2) + 5(-3)) = 0

0 = 0

So therefore this is closed under scalar multiplication.

I think I'm correct in that, but how do I find out addition?