Determine whether the set $\mathbb{R}^2$, with the operations $(x\_1,y\_1)+(x\_2,y\_2)=(x\_1x\_2,y\_1y\_2)$ and $c(x\_1,y\_1)=(cx\_1,cy\_1)$, is a vector space. If it is, verify each vector space axiom; if not, state all vector space axioms that fail.

I know this set is not a vector space under the specified operations since axioms like distributive multiplication do not hold true, but my problem is that the book uses the below argument to deduce that it's not a vector space:

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> "The set is not a vector space. It fails because $(1,1)$ is the additive identity, so $(0,0)$ has no additive inverse."
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If we pick vector $u=(0,0)$ to be in the set $\mathbb{R}^2$, then and according to the book's solution, $u$ has no additive inverse. Why is this the case? I mean isn't $(0,0)$ in this question the additive inverse of all vectors in the set $\mathbb{R}^2$ including $u$? Is this related to

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> "The additive inverse of $u$ is unique. That is, if $u+v=0$, then $v =-u$."
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And what is meant by "unique" in the previous statement?