The complex numbers \(z , w\) are given by \(z = 3 - 4 \mathrm { i } , w = 2 - \mathrm { i }\).
(i) Find the modulus and argument of \(z w\).
(ii) Express \(z w\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\).
The complex numbers \(v , w , z\) satisfy the equation \(\frac { 1 } { v } = \frac { 1 } { w } - \frac { 1 } { z }\). Find \(v\) in the form \(a + \mathrm { i } b\), where \(a , b\) are real.
The complex conjugate of \(v\) is denoted by \(\bar { v }\).
Show that \(v \bar { v } = k\), where \(k\) is a real number whose value is to be determined.