- With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations
$$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r }
12
- 4
5
\end{array} \right) + \lambda \left( \begin{array} { r }
5
- 4
2
\end{array} \right) , \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { l }
2
2
0
\end{array} \right) + \mu \left( \begin{array} { l }
0
6
3
\end{array} \right)$$
where \(\lambda\) and \(\mu\) are scalar parameters.
- Show that \(l _ { 1 }\) and \(l _ { 2 }\) meet, and find the position vector of their point of intersection \(A\).
- Find, to the nearest \(0.1 ^ { \circ }\), the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\)
The point \(B\) has position vector \(\left( \begin{array} { l } 7
0
3 \end{array} \right)\). - Show that \(B\) lies on \(l _ { 1 }\)
- Find the shortest distance from \(B\) to the line \(l _ { 2 }\), giving your answer to 3 significant figures.