$W$ 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!

No! it is not closed under scalar multiplication. For example let $\lambda=1/2$ and $(x,y,z)\in W$ then $\lambda (x,y,z)\notin W$ and when $\lambda <0$ it is obvious that $W$ is not closed.
