(a) Express the complex number \(w = 4 \sqrt { 3 } - 4 \mathrm { i }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\) where \(r > 0\) and \(- \pi < \theta \leqslant \pi\)
(b) Show, on a single Argand diagram,
the point representing \(w\)
the locus of points defined by \(\arg ( z + 10 i ) = \frac { \pi } { 3 }\)
(c) Hence determine the minimum distance of \(w\) from the locus \(\arg ( z + 10 i ) = \frac { \pi } { 3 }\)