Show lines are skew (non-intersecting)

10 With respect to the origin \(O\), the points \(A\) and \(B\) have position vectors given by $$\overrightarrow { O A } = 2 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = \mathbf { i } + 4 \mathbf { j } + 3 \mathbf { k }$$ The line \(l\) has vector equation \(\mathbf { r } = 4 \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } + s ( \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\).

1. Prove that the line \(I\) does not intersect the line through \(A\) and \(B\).
2. Find the equation of the plane containing \(l\) and the point \(A\), giving your answer in the form \(a x + b y + c z = d\). \footnotetext{Every reasonable effort has been made to trace all copyright holders where the publishers (i.e. UCLES) are aware that third-party material has been reproduced. The publishers would be pleased to hear from anyone whose rights they have unwittingly infringed.
   University of Cambridge International Examinations is part of the University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

8 The points \(A\) and \(B\) have position vectors, relative to the origin \(O\), given by \(\overrightarrow { O A } = \mathbf { i } + \mathbf { j } + \mathbf { k }\) and \(\overrightarrow { O B } = 2 \mathbf { i } + 3 \mathbf { k }\). The line \(l\) has vector equation \(\mathbf { r } = 2 \mathbf { i } - 2 \mathbf { j } - \mathbf { k } + \mu ( - \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\).

1. Show that the line passing through \(A\) and \(B\) does not intersect \(l\).
2. Show that the length of the perpendicular from \(A\) to \(l\) is \(\frac { 1 } { \sqrt { 2 } }\).

8 With respect to the origin \(O\), the position vectors of the points \(A , B , C\) and \(D\) are given by $$\overrightarrow { O A } = \left( \begin{array} { l } 2 \\ 1 \\ 5 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 4 \\ - 1 \\ 1 \end{array} \right) , \quad \overrightarrow { O C } = \left( \begin{array} { l } 1 \\ 1 \\ 2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O D } = \left( \begin{array} { l } 3 \\ 2 \\ 3 \end{array} \right)$$

1. Show that \(A B = 2 C D\).
2. Find the angle between the directions of \(\overrightarrow { A B }\) and \(\overrightarrow { C D }\).
3. Show that the line through \(A\) and \(B\) does not intersect the line through \(C\) and \(D\).