I'm having trouble interpretation some equations.

To prove whether a given set is a vector space, I know there are ~10 axioms I need to test, but I'm having trouble interpreting the equations to be able to test the axioms.

Here is one of the equations:

$$V=\big\{x=\begin{pmatrix} x\_1 \\ x\_2 \\ x\_3 \end{pmatrix}\in \mathbb{R}^3;x\_1+x\_2+x\_3=0 \big\}$$

How do I apply the axioms to this set. Any hints would be great.

EDIT: I think this is a subset of $\mathbb{R}^3$ if I'm reading this correctly, so I guess I should be able to just test if this is a subset.
This would mean I need to test:

1. The zero vector of $\mathbb{R}^3$ is in V
2. V is closed under addition
3. H is closed under multiplication by scalars

Is this the right way to go about this? Would I just plug in for the $x\_n$ terms?