Let $z _ { 1 } , \ldots , z _ { n }$ be complex numbers. Show that if
$$\left| z _ { 1 } + \cdots + z _ { n } \right| = \left| z _ { 1 } \right| + \cdots + \left| z _ { n } \right|$$
then the vector $\left( \begin{array} { c } z _ { 1 } \\ \vdots \\ z _ { n } \end{array} \right)$ is collinear with the vector $\left( \begin{array} { c } \left| z _ { 1 } \right| \\ \vdots \\ \left| z _ { n } \right| \end{array} \right)$.