Is the function $d(x,y) = \frac{\|x-y\|}{\|x\|\|y\|}$ a metric?
Solution 1:
$\newcommand{\Reals}{\mathbf{R}}$Identify the set of non-zero vectors in $\Reals^{2}$ with the set of non-zero complex numbers. The Euclidean norm corresponds with the complex modulus, so if $x$ and $y$ are non-zero, then $$ \frac{\|x - y\|}{\|x\|\, \|y\|} = \left\|\frac{x - y}{xy}\right\| = \left\|\frac{1}{y} - \frac{1}{x}\right\|. $$ That is, $d$ corresponds to the ordinary Euclidean distance after a bijection (the complex reciprocal map), and therefore satisfies the triangle inequality.