8 The complex numbers \(u\) and \(v\) are defined by \(u = - 4 + 2 \mathrm { i }\) and \(v = 3 + \mathrm { i }\).
- Find \(\frac { u } { v }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
- Hence express \(\frac { u } { v }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r\) and \(\theta\) are exact.
In an Argand diagram, with origin \(O\), the points \(A , B\) and \(C\) represent the complex numbers \(u , v\) and \(2 u + v\) respectively. - State fully the geometrical relationship between \(O A\) and \(B C\).
- Prove that angle \(A O B = \frac { 3 } { 4 } \pi\).