1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively.]

An aeroplane makes a journey from a point \(P\) to a point \(Q\) which is due east of \(P\). The wind velocity is \(w ( \cos \theta \mathbf { i } + \sin \theta \mathbf { j } )\), where \(w\) is a positive constant. The velocity of the aeroplane relative to the wind is \(v ( \cos \phi \mathbf { i } - \sin \phi \mathbf { j } )\), where \(v\) is a constant and \(v > w\). Given that \(\theta\) and \(\phi\) are both acute angles,

1. show that \(v \sin \phi = w \sin \theta\),
2. find, in terms of \(v , w\) and \(\theta\), the speed of the aeroplane relative to the ground.