This is not true. For example, if \(y=0\), and \(\displaystyle{x}≠{0}\) is some non-trivial vector, then \(\displaystyle{\left|{\left|{x}+{y}\right|}\right|}={\left|{\left|{x}+{0}\right|}\right|}={\left|{\left|{x}\right|}\right|}={\left|{\left|{x}\right|}\right|}+{\left|{\left|{y}\right|}\right|}\rbrace={0}\)

Now, if \(x=cy\), then \(x=0\), which is impossible. So suppose that there exists \(c>0\) such that \(y=cx\). However, \(||cx||=|c||x||>0\), so \(\displaystyle{c}{x}≠{0}\), but \(y=0\). Therefore, there exists no \(c>0\) such that \(x=cy\ or\ y=cx\).