4 The cube roots of 1 are denoted by \(1 , \omega\) and \(\omega ^ { 2 }\), where the imaginary part of \(\omega\) is positive.
- Show that \(1 + \omega + \omega ^ { 2 } = 0\).
\includegraphics[max width=\textwidth, alt={}, center]{d12573dd-c0c2-4f0d-8e49-8fdf8d5864a5-2_616_748_1676_699}
In the diagram, \(A B C\) is an equilateral triangle, labelled anticlockwise. The points \(A , B\) and \(C\) represent the complex numbers \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) respectively. - State the geometrical effect of multiplication by \(\omega\) and hence explain why \(z _ { 1 } - z _ { 3 } = \omega \left( z _ { 3 } - z _ { 2 } \right)\).
- Hence show that \(z _ { 1 } + \omega z _ { 2 } + \omega ^ { 2 } z _ { 3 } = 0\).