Going off the axioms, I'm trying to determine how to verify each of these. In the first set, do I take $u = (x_1, x_2, x_3)$ and $v = (y_1, y_2, y_3)$, or would it rather be $u = (x_1 + y_1, x_2 + y_2, x_3 + y_3), v = (x_4 + y_4, x_5 + y_5, x_6 + y_6)$?
Based on what I know about scalar multiplication, I would say the first set is not a vector space because $0(x_1, x_2, x_3)$ = $(0, 0, 0)$ which is not equivalent to $(0x_1, x_2, x_3)$, for non-zero values of $x_2$ and $x_3$. The second and third sets seem to define vector spaces, as I can't seem to figure out any axiom that directly fails (i.e. $u+v$ is in $V$, a unique additive identity exists, and the set is closed under scalar multiplication). Any help would be appreciated.
Edit: The third set is not a vector space.
The third one is not a vector space. The zero vector in this space is $(-1,-1)$ but if you multiply this by $2$ you don't get back the zero vector.