Express \(w\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
The complex number \(1 + 2 \mathrm { i }\) is denoted by \(u\). The complex number \(v\) is such that \(| v | = 2 | u |\) and \(\arg v = \arg u + \frac { 1 } { 3 } \pi\).
Sketch an Argand diagram showing the points representing \(u\) and \(v\).
Explain why \(v\) can be expressed as \(2 u w\). Hence find \(v\), giving your answer in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real and exact.