How do I determine if the set of triples of all real numbers on $\mathbb{R}$ is a vector space.

How do I determine if the set of triples of all real numbers on
$\mathbb{R}$ is a vector space.

I need to confirm if the set of all triples of real numbers of the form
$(0, y, z)$ where $y=z$ with standard operations of addition and scalar
multiplication on $\mathbb{R}^3$ is a vector space. Any clues will be
greatly appreciates.