(i) The complex number \(w\) has modulus 1 and argument \(\frac{1}{3} \pi\). Therefore, \(w = \cos \frac{1}{3} \pi + i \sin \frac{1}{3} \pi\).

Calculating, \(\cos \frac{1}{3} \pi = \frac{1}{2}\) and \(\sin \frac{1}{3} \pi = \frac{\sqrt{3}}{2}\).

Thus, \(w = \frac{1}{2} + \frac{\sqrt{3}}{2}i\).

(ii) On the Argand diagram, plot \(u = 1 + 2i\) and \(v\) such that \(|v| = 2|u|\) and \(\arg v = \arg u + \frac{1}{3} \pi\).

(iii) Since \(|v| = 2|u|\), \(v\) can be expressed as \(2uw\) because \(|w| = 1\) and \(\arg v = \arg u + \arg w\).

Calculate \(2uw = 2(1 + 2i)\left(\frac{1}{2} + \frac{\sqrt{3}}{2}i\right)\).

First, find \((1 + 2i)\left(\frac{1}{2} + \frac{\sqrt{3}}{2}i\right) = \frac{1}{2} + \frac{\sqrt{3}}{2}i + i + \sqrt{3}i^2\).

Since \(i^2 = -1\), this becomes \(\frac{1}{2} + \frac{\sqrt{3}}{2}i + i - \sqrt{3}\).

Simplify to \(\left(\frac{1}{2} - \sqrt{3}\right) + \left(\frac{\sqrt{3}}{2} + 1\right)i\).

Multiply by 2: \(2uw = 1 - 2\sqrt{3} + (2 + \sqrt{3})i\).