Q1 (a) The points \( A, B, C \) and \( D \) have Cartesian coordinates \( (1,5,4),(3,7,1),(3,2,3) \) and \( (2,5,5) \) respectively.
(i) Find the parametric equations of two lines, one passing through \( A \) and \( B \), and the other passing through \( C \) and \( D \).
(ii) Find the shortest distance between these lines.
(iii) Find the volume of a parallelepiped with edges \( A B, A C \) and \( A D \).
(b) In \( \mathbb{R}^{3} \) the spherical polar basis vectors are
\[
\begin{array}{l}
\mathbf{e}_{r}=\sin \theta \cos \phi \mathbf{i}+\sin \theta \sin \phi \mathbf{j}+\cos \theta \mathbf{k} \\
\mathbf{e}_{\theta}=\cos \theta \cos \phi \mathbf{i}+\cos \theta \sin \phi \mathbf{j}-\sin \theta \mathbf{k} \\
\mathbf{e}_{\phi}=-\sin \phi \mathbf{i}+\cos \phi \mathbf{j}
\end{array}
\]
(i) What does it mean for a set of vectors to be orthonormal?
(ii) Show that the set of spherical polar basis vectors is orthonormal.
(iii) In spherical polar coordinates \( (r, \theta, \phi) \) the angles \( \theta \) and \( \phi \) are functions of time. Find expressions for \( \dot{\mathbf{e}}_{r}, \dot{\mathbf{e}}_{\theta} \) and \( \dot{\mathbf{e}}_{\phi} \) in terms of spherical polar coordinates and spherical polar basis vectors.
