A ship is steaming due east at a constant speed. The ship sends out an SOS call which is received by an aeroplane. The navigator of the aeroplane correctly determines the bearing of the ship as \(\alpha\) radians east of north, and calculates that, allowing for the motion of the ship, if they fly on a bearing \(\beta\) radians east of north at speed \(v\), they should reach the ship after flying a distance \(l\). The pilot accepts this course, but due to errors in his instruments he actually flies on a bearing \((\beta + \phi)\) radians east of north at speed \(v(1 + \epsilon)\), where \(\phi\) and \(\epsilon\) are small. Show that, to first order in \(\phi\) and \(\epsilon\), their closest distance of approach to the ship is $$l[\epsilon \sin(\beta - \alpha) + \phi \cos(\beta - \alpha)].$$
\(P\), \(Q\), \(O\) and \(R\) are four distinct points which are not coplanar. Let \(a\) be the angle between the planes \(QOP\) and \(POR\). Define \(b\), \(c\), \(A\), \(B\) and \(C\) similarly by cyclic permutation of \(P\), \(Q\) and \(R\). Let \(\mathbf{p}\), \(\mathbf{q}\) and \(\mathbf{r}\) be the position vectors of \(P\), \(Q\) and \(R\) respectively with respect to \(O\) as origin, and let \(p\), \(q\), and \(r\) be the corresponding magnitudes of these vectors. By geometrical considerations, evaluate $$|(\mathbf{p} \times \mathbf{q}) \times (\mathbf{p} \times \mathbf{r})|$$ in terms of \(p\), \(q\), \(r\) and the angles \(a\), \(b\), \(c\), \(A\), \(B\) and \(C\). By expanding this repeated vector product, show also that $$\frac{|(\mathbf{p} \times \mathbf{q}) \times (\mathbf{p} \times \mathbf{r})|}{|(\mathbf{q} \times \mathbf{r}) \times (\mathbf{q} \times \mathbf{p})|} = \frac{p}{q}.$$ Deduce that $$\frac{\sin a}{\sin A} = \frac{\sin b}{\sin B} = \frac{\sin c}{\sin C}.$$ [It may be assumed without proof that, for any three vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\), $$(\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{b} \cdot \mathbf{c})\mathbf{a}$$ and $$(\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} = (\mathbf{b} \times \mathbf{c}) \cdot \mathbf{a} = (\mathbf{c} \times \mathbf{a}) \cdot \mathbf{b}.]$$