Showing that this set satisfies the closed criterion

Suppose we have the subset $S = \{ \lambda v \mid \lambda \geq 0 \} + K$, where $v$ is a vector in $\mathbb{R^3}$ and $K$ is a convex hull of six other vectors.

How do I show that it satisfies the definition of being a closed set and the closure condition?

My idea is to see if the vectors are independent linear combinations and then take two points of the given six vectors and see if it's in the convex hull. Am I right? If not, how do I proceed?

(a) The ray $\{\lambda v | \lambda \ge 0\}$ is closed. This is rather easy. Nothing to do here.
Conclude that your set $S$ is closed.