For points $x,y\in\mathbb{R}^{2}$ define $d(x,y)=|(x_{1}-y_{2})^{2}-(x_{2}-y_{1})^{2}|$. Does $d$ define a metric space?

Question

So I deduced that $d(x,y)$ is symmetric and reflexive fairly trivially, but I can not figure out how to prove whether or not $d(x,y)+d(y,z)\geq d(x,z)$ holds. I first tried to prove that it did not hold by plugging in random points, but then I did not know how to proceed. Any advice would be greatly appreciated. Thank you.

Answer 1

No, it's not a metric space. In the "reflexivity" axiom, you need to check that $d(x,y)=0$ if and only if $x=y$. But here, $d((1,2), \, (2,1))=0$.

The triangle inequality is also false. Consider for example $x = (1,0)$, $y = (0,0)$ and $z = (2,2)$. Then $d(x,z) = 3$ where $d(x,y)+d(y,z) = 1+0 = 1$.