Line intersection (vector form)

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

1. 1. Show that the two lines intersect.
      [0pt] [3 marks]
      15
2. (ii) Find the position vector of the point of intersection.
   15
3. 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
4. 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 \(0.1 ^ { \circ }\)

7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left[ \begin{array} { c } 3 \\ 1 \\ - 2 \end{array} \right] + \lambda \left[ \begin{array} { c } 3 \\ - 4 \\ 1 \end{array} \right] \\ & l _ { 2 } : \mathbf { r } = \left[ \begin{array} { c } - 12 \\ a \\ - 3 \end{array} \right] + \mu \left[ \begin{array} { c } 3 \\ 2 \\ - 1 \end{array} \right] \end{aligned}$$ 7

1. Show that the point \(P ( - 3,9 , - 4 )\) lies on \(l _ { 1 }\) 7
2. Show that \(l _ { 1 }\) is perpendicular to \(l _ { 2 }\) 7
3. Given that the lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect, calculate the value of the constant \(a\) 7
4. Hence, find the coordinates of the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\)

2 The lines \(L _ { 1 }\) and \(L _ { 2 }\) have the following equations. \(L _ { 1 } : \mathbf { r } = \left( \begin{array} { c } - 5 \\ 6 \\ 15 \end{array} \right) + \lambda \left( \begin{array} { c } 5 \\ - 2 \\ - 2 \end{array} \right)\) \(L _ { 2 } : \mathbf { r } = \left( \begin{array} { c } 24 \\ 1 \\ - 5 \end{array} \right) + \mu \left( \begin{array} { c } 3 \\ 1 \\ - 4 \end{array} \right)\)

1. Show that \(L _ { 1 }\) and \(L _ { 2 }\) intersect, giving the position vector of the point of intersection.
2. Find the equation of the line which intersects \(L _ { 1 }\) and \(L _ { 2 }\) and is perpendicular to both. Give your answer in cartesian form.

1 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have the following equations. $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 8 \\ - 11 \\ - 2 \end{array} \right) + \lambda \left( \begin{array} { r } - 2 \\ 5 \\ 3 \end{array} \right) \\ & l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } - 6 \\ 11 \\ 8 \end{array} \right) + \mu \left( \begin{array} { r } - 3 \\ 1 \\ - 1 \end{array} \right) \end{aligned}$$

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
2. Write down the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).