15 Two submarines are travelling on different straight lines.
The two lines are described by the equations
$$\mathbf { r } = \left[ \begin{array} { c }
2
- 1
4
\end{array} \right] + \lambda \left[ \begin{array} { c }
5
3
- 2
\end{array} \right] \quad \text { and } \quad \frac { x - 5 } { 4 } = \frac { y } { 2 } = 4 - z$$
15
- Show that the two lines intersect.
[0pt]
[3 marks]
15
- (ii) Find the position vector of the point of intersection.
15 - Tracey says that the submarines will collide because there is a common point on the two lines.
Explain why Tracey is not necessarily correct.
15
- Calculate the acute angle between the lines
$$\mathbf { r } = \left[ \begin{array} { c }
2
- 1
4
\end{array} \right] + \lambda \left[ \begin{array} { c }
5
3
- 2
\end{array} \right] \quad \text { and } \quad \frac { x - 5 } { 4 } = \frac { y } { 2 } = 4 - z$$
Give your angle to the nearest \(0.1 ^ { \circ }\)