Is this a vectorial space

Question

I can't understand why the 0 vector here is not unique?

Answer 1

$V$ is not a vector space because it does not respect the commutativity and associativity of addition.

There is an unique zero vector $(0,0)$.

Answer 2

Hint

If $(V,+,\cdot)$ is a vector space then $(V,+)$ is an abelian group. Can you give a counterexample to show that it isn't the case?

Answer 3

We want a zero vector to satisfy $0+u=x+0=u$ for all vectors $u$. However there is no such vector. Suppose $0=(a,b)$. Then $0+(x,y)=(a+2x,b+3y)=(x,y)$. But then $a+2x=x, b+3y=y$ for all $x,y$, which is impossible.