Line intersection with line

1. Two birds are flying towards their nest, which is in a tree.

Relative to a fixed origin, the flight path of each bird is modelled by a straight line.
In the model, the equation for the flight path of the first bird is $$\mathbf { r } _ { 1 } = \left( \begin{array} { r } - 1 \\ 5 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { l } 2 \\ a \\ 0 \end{array} \right)$$ and the equation for the flight path of the second bird is $$\mathbf { r } _ { 2 } = \left( \begin{array} { r } 4 \\ - 1 \\ 3 \end{array} \right) + \mu \left( \begin{array} { r } 0 \\ 1 \\ - 1 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(a\) is a constant.
In the model, the angle between the birds' flight paths is \(120 ^ { \circ }\)

1. Determine the value of \(a\).
2. Verify that, according to the model, there is a common point on the flight paths of the two birds and find the coordinates of this common point. The position of the nest is modelled as being at this common point.
   The tree containing the nest is in a park.
   The ground level of the park is modelled by the plane with equation $$2 x - 3 y + z = 2$$
3. Hence determine the shortest distance from the nest to the ground level of the park.
4. By considering the model, comment on whether your answer to part (c) is reliable, giving a reason for your answer.

3
1 \end{array} \right) + \lambda \left( \begin{array} { r } - 2
4
- 2 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l }

$$\begin{aligned} l _ { 1 } : \mathbf { r } = \left( \begin{array} { c } 4 \\ 2 \\ 5 \end{array} \right) + \lambda \left( \begin{array} { c } 1 \\ 3 \\ - 2 \end{array} \right) \\ & l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 1 \\ a \\ 1 \end{array} \right) + \mu \left( \begin{array} { c } 2 \\ 2 \\ - 3 \end{array} \right) \end{aligned}$$

1. Find the position vector of the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).
2. Determine the value of \(a\).

10 Lines \(L _ { 1 } , L _ { 2 }\) and \(L _ { 3 }\) have vector equations $$\begin{aligned} & L _ { 1 } = ( 4 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) + s ( 6 \mathbf { i } + 9 \mathbf { j } - 3 \mathbf { k } ) , \\ & L _ { 2 } = ( 2 \mathbf { i } + 3 \mathbf { j } ) + t ( - 3 \mathbf { i } - 8 \mathbf { j } + 6 \mathbf { k } ) , \\ & L _ { 3 } = ( 3 \mathbf { i } - \mathbf { j } + 4 \mathbf { k } ) + u ( - 2 \mathbf { i } + c \mathbf { j } + \mathbf { k } ) . \end{aligned}$$ In each of the following cases, find the value of \(c\).

1. \(\quad L _ { 1 }\) and \(L _ { 3 }\) are parallel.
2. \(\quad L _ { 2 }\) and \(L _ { 3 }\) intersect.

8

1. Show that the lines $$\mathbf { r } = - 3 \mathbf { i } + \mathbf { j } - 5 \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } + \mathbf { 6 } \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 4 \mathbf { i } + \mathbf { j } + 5 \mathbf { k } + \mu ( - 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } )$$ intersect and write down the coordinates of their point of intersection.
2. Find in degrees the obtuse angle between the two lines.

With respect to a fixed origin \(O\), the equations of lines \(l_1\) and \(l_2\) are given by $$l_1: \mathbf{r} = \begin{pmatrix} 2 \\ 8 \\ 10 \end{pmatrix} + \lambda \begin{pmatrix} -1 \\ 2 \\ 3 \end{pmatrix}$$ $$l_2: \mathbf{r} = \begin{pmatrix} -4 \\ -1 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} 5 \\ 4 \\ 8 \end{pmatrix}$$ where \(\lambda\) and \(\mu\) are scalar parameters. Prove that lines \(l_1\) and \(l_2\) are skew. [5]

Line \(l_1\) has equation $$\frac{x - 2}{3} = \frac{1 - 2y}{4} = -z$$ and line \(l_2\) has equation $$\mathbf{r} = \begin{bmatrix} -7 \\ 4 \\ -2 \end{bmatrix} + \mu \begin{bmatrix} 12 \\ a + 3 \\ 2b \end{bmatrix}$$

1. In the case when \(l_1\) and \(l_2\) are parallel, show that \(a = -11\) and find the value of \(b\). [4 marks]
2. In a different case, the lines \(l_1\) and \(l_2\) intersect at exactly one point, and the value of \(b\) is 3 Find the value of \(a\). [5 marks]

The equations of two intersecting lines are $$\mathbf{r} = \begin{pmatrix} -12 \\ a \\ -1 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 2 \\ 1 \end{pmatrix} \quad \mathbf{r} = \begin{pmatrix} 2 \\ 0 \\ 5 \end{pmatrix} + \mu \begin{pmatrix} -3 \\ 1 \\ -1 \end{pmatrix}$$ where \(a\) is a constant.

1. Find a vector, \(\mathbf{b}\), which is perpendicular to both lines. [2]
2. Show that \(\mathbf{b} \cdot \begin{pmatrix} -12 \\ a \\ -1 \end{pmatrix} = \mathbf{b} \cdot \begin{pmatrix} 2 \\ 0 \\ 5 \end{pmatrix}\). [2]
3. Hence, or otherwise, find the value of \(a\). [2]

Line \(l_1\) has Cartesian equation $$l_1: \frac{-x}{2} = \frac{y-5}{2} = \frac{-z-6}{7}.$$

1. Find a vector equation for \(l_1\). [2]

Line \(l_2\) has vector equation $$l_2: \mathbf{r} = \begin{pmatrix} 2 \\ 7 \\ -1 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ -2 \\ 4 \end{pmatrix}.$$

1. Find the point of intersection of \(l_1\) and \(l_2\). [3]
2. Find the acute angle between \(l_1\) and \(l_2\). [3]