7 The complex numbers \(- 2 + \mathrm { i }\) and \(3 + \mathrm { i }\) are denoted by \(u\) and \(v\) respectively.
- Find, in the form \(x + \mathrm { i } y\), the complex numbers
(a) \(u + v\),
(b) \(\frac { u } { v }\), showing all your working. - State the argument of \(\frac { u } { v }\).
In an Argand diagram with origin \(O\), the points \(A , B\) and \(C\) represent the complex numbers \(u , v\) and \(u + v\) respectively.
- Prove that angle \(A O B = \frac { 3 } { 4 } \pi\).
- State fully the geometrical relationship between the line segments \(O A\) and \(B C\).