Vector space question need help

Im stuck on this question does anyone know how to do it i have no idea :(

Determine if the set of all triples of real numbers of the form (0,y,z) where y=z with standard operations on R3 is a vector space.

Thankyou if you can help

first the set is not empty (0,5,5) is an element of the set

if you multiply any element of the set by some number
m(0,y,y)=(0,my,my) so the same type of triple
it belongs to set

and the last, sum of any two elements of the set belongs to set
(0,y,y)+(0,t,t)=(0,y+t,y+t)

belongs to set.
so this set is vector space with standard operations.