$W$ is a subset of $\mathbb{R}^3$ defined as $W = \{(x, y, z)|x + y + z ≥ 1\}$. Is $W$ closed under vector addition and scalar multiplication? $W$ is a subset of $\mathbb{R}^3$ defined as  $W = \{(x, y, z)|x + y + z ≥ 1\}$.
(a) Is $W$ closed under vector addition?
If your answer is no, then find two vectors $u, v \in W$ such that $u + v \not\in W$.
(b) Is $W$ closed under scalar multiplication?
If your answer is no, then find a vector $u \in W$ and a scalar $c$ such that $cu \not\in W$.
(c) Is $W$ a subspace of $\mathbb{R}^3$?
I know the answer for a is yes and b is no and c is no
But I do not understand how to get there I have been researching it for a while, but I still do not understand, please help!

Hints:
a) Let $u,v\in W$ where $u=(u_1,u_2,u_3)$ and $v=(v_1,v_2,v_3)$. Now we have $u+v=(u_1+v_1,u_2+v_2,u_3+v_3)$. Now what can we say about 
$$(u_1+v_1)+(u_2+v_2)+(u_3+v_3)$$
given that we know $u_1+u_2+u_3\ge1$ and $v_1+v_2+v_3\ge1$?
b) Scalars are just defined as any real number. What happens when we choose $c=0$?
c) Subspaces must be closed under addition and scalar multiplication. You can argue that $W$ is not closed under one of these things from parts a) and b).
