Is the relation $x+3y = 0$ antisymmetric for all real numbers? I don't think so, because it is never $aRb$ nor $bRa$ and it is never $aRa$ or $bRb$, thus it is always false, but I don't know if I understood what antisymmetric means exactly.
 A: 
Given the relation $\, \mathcal{R}$ defined on $\mathbb{R}$ such that $x\,\mathcal{R}\, y\,$ if and only if $x + 3y = 0$: 

$\,\mathcal{R}\,$ is antisymmetric if and only if, for all $a, b\in \mathbb{R},$ whenever both $ a\,\mathcal{R}\, b\,$ and $b \mathcal{R} a$, then it must be the case that $a=b$. 
Let $a, b\in \mathbb{R}$ and define $R$ such that $a \,R \,b$ if and only if $a + 3b = 0$.
$$a\,R\, b \;\text{ and}\;\; b \,R \,a \implies a + 3 b = 0 \,\text{ and}\;\, b + 3a = 0.$$ 
$$\iff a + 3b = b + 3a$$
$$\iff -2a = -2b$$
$$\iff a=b.$$
Therefore, $\forall a, b \in \mathbb{R},\;a \, R \, b \;\text{ and}\;\; b \,R \,a \implies a=b.$ 
So the relation IS antisymmetric on $\mathbb{R}$ for all real numbers since for $a, b \in \mathbb{R}$ IF both $\,a \,\mathcal{R} \,b\,$ and $\,b\, \mathcal{R} \,a$, then $a = b$.
The fact that there are no pairs $a, b \in \mathbb{R}$ other than $a = b = 0$ where both $\,a\, \mathcal{R} \,b\,$ and $\,b \,\mathcal{R} \,a\,$ doesn't matter. All that matters when determining whether a relation is antisymmtric, is that  if and when it happens to be the case that there are $a, b$ such that both $\,a\,\mathcal{R} \,b\,$ AND $\,b\,\mathcal{R}\, a\,$ then it must follow that $a=b$. 
In short, the relation is antisymmetric.

Edit, added for clarification in response to comments:
Note that antisymmetric is not the opposite of symmetric (the term is misleading in that sense). 
