Problems

The line \(l\) has vector equation \({\bf r} = \lambda {\bf s}\), where \[ {\bf s} = (\cos\theta+\sqrt3\,) \; {\bf i} +(\surd2\;\sin\theta)\;{\bf j} +(\cos\theta-\sqrt3\,)\;{\bf k} \] and \(\lambda\) is a scalar parameter. Find an expression for the angle between \(l\) and the line \mbox{\({\bf r} = \mu(a\, {\bf i} + b\,{\bf j} +c\, {\bf k})\)}. Show that there is a line \(m\) through the origin such that, whatever the value of \(\theta\), the acute angle between \(l\) and \(m\) is \(\pi/6\). A plane has equation \(x-z=4\sqrt3\). The line \(l\) meets this plane at \(P\). Show that, as \(\theta\) varies, \(P\) describes a circle, with its centre on \(m\). Find the radius of this circle.

A straight stick of length \(h\) stands vertically. On a sunny day, the stick casts a shadow on flat horizontal ground. In cartesian axes based on the centre of the Earth, the position of the Sun may be taken to be \(R(\cos\theta,\sin\theta,0)\) where \(\theta\) varies but \(R\) is constant. The positions of the base and tip of the stick are \(a(0,\cos\phi,\sin\phi)\) and \(b(0,\cos\phi,\sin\phi)\), respectively, where \(b-a=h\). Show that the displacement vector from the base of the stick to the tip of the shadow is \[ Rh(R\cos\phi\sin\theta-b)^{-1}\begin{pmatrix}-\cos\theta\\ -\sin^{2}\phi\sin\theta\\ \cos\phi\sin\phi\sin\theta \end{pmatrix}. \] {[}`Stands vertically' means that the centre of the Earth, the base of the stick and the tip of the stick are collinear, `horizontal' means perpendicular to the stick.