4.04e Line intersections: parallel, skew, or intersecting

\(\mathbf { 7 } \quad\) The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left[ \begin{array} { r } 0 \\ - 2 \\ q \end{array} \right] + \lambda \left[ \begin{array} { r } 2 \\ 0 \\ - 1 \end{array} \right]\), where \(q\) is an integer. The line \(l _ { 2 }\) has equation \(\mathbf { r } = \left[ \begin{array} { l } 8 \\ 3 \\ 5 \end{array} \right] + \mu \left[ \begin{array} { l } 2 \\ 5 \\ 4 \end{array} \right]\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(P\).

1. Show that \(q = 4\) and find the coordinates of \(P\).
2. Show that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular.
3. The point \(A\) lies on the line \(l _ { 1 }\) where \(\lambda = 1\).
   1. Find \(A P ^ { 2 }\).
   2. The point \(B\) lies on the line \(l _ { 2 }\) so that the right-angled triangle \(A P B\) is isosceles. Find the coordinates of the two possible positions of \(B\).

9. The equations of the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by $$\begin{array} { l l } l _ { 1 } : & \mathbf { r } = \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) , \\ l _ { 2 } : & \mathbf { r } = - 2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k } + \mu ( 2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } ) , \end{array}$$ where \(\lambda\) and \(\mu\) are parameters.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the coordinates of \(Q\), their point of intersection.
2. Show that \(l _ { 1 }\) is perpendicular to \(l _ { 2 }\). The point \(P\) with \(x\)-coordinate 3 lies on the line \(l _ { 1 }\) and the point \(R\) with \(x\)-coordinate 4 lies on the line \(l _ { 2 }\).
3. Find, in its simplest form, the exact area of the triangle \(P Q R\). END

5. A straight road passes through villages at the points \(A\) and \(B\) with position vectors ( \(9 \mathbf { i } - 8 \mathbf { j } + 2 \mathbf { k }\) ) and ( \(4 \mathbf { j } + \mathbf { k }\) ) respectively, relative to a fixed origin. The road ends at a junction at the point \(C\) with another straight road which lies along the line with equation $$\mathbf { r } = ( 2 \mathbf { i } + 16 \mathbf { j } - \mathbf { k } ) + \mu ( - 5 \mathbf { i } + 3 \mathbf { j } ) ,$$ where \(\mu\) is a scalar parameter.

1. Find the position vector of \(C\). Given that 1 unit on each coordinate axis represents 200 metres,
2. find the distance, in kilometres, from the village at \(A\) to the junction at \(C\).
   5. continued

8

1. Given that the lines \(\mathbf { r } = \left( \begin{array} { l } 0 \\ 2 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ 1 \\ 3 \end{array} \right)\) and \(\mathbf { r } = \left( \begin{array} { r } - 1 \\ 2 \\ k \end{array} \right) + \mu \left( \begin{array} { l } 2 \\ 3 \\ 4 \end{array} \right)\) meet, determine \(k\).
2. In this question you must show detailed reasoning. Find the acute angle between the two lines.

1. The line \(l\) has equation

$$\frac { x + 2 } { 1 } = \frac { y - 5 } { - 1 } = \frac { z - 4 } { - 3 }$$ The plane \(\Pi\) has equation $$\mathbf { r } . ( \mathbf { i } - 2 \mathbf { j } + \mathbf { k } ) = - 7$$ Determine whether the line \(l\) intersects \(\Pi\) at a single point, or lies in \(\Pi\), or is parallel to \(\Pi\) without intersecting it.
(5)

1. The line \(l _ { 1 }\) has equation

$$\frac { x - 1 } { 2 } = \frac { y + 1 } { - 1 } = \frac { z - 4 } { 3 }$$ The line \(l _ { 2 }\) has equation $$\mathbf { r } = \mathbf { i } + 3 \mathbf { k } + t ( \mathbf { i } - \mathbf { j } + 2 \mathbf { k } )$$ where \(t\) is a scalar parameter.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) lie in the same plane.
2. Write down a vector equation for the plane containing \(l _ { 1 }\) and \(l _ { 2 }\)
3. Find, to the nearest degree, the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\)

5 The line \(l _ { 1 }\) has equation \(\frac { x + 5 } { 1 } = \frac { y + 4 } { - 3 } = \frac { z - 3 } { 5 }\) The plane \(\Pi _ { 1 }\) has equation \(2 x + 3 y - 2 z = 6\)

1. Find the point of intersection of \(l _ { 1 }\) and \(\Pi _ { 1 }\) The line \(l _ { 2 }\) is the reflection of the line \(l _ { 1 }\) in the plane \(\Pi _ { 1 }\)
2. Show that a vector equation for the line \(l _ { 2 }\) is $$\mathbf { r } = \left( \begin{array} { r } - 7 \\ 2 \\ - 7 \end{array} \right) + \mu \left( \begin{array} { c } 10 \\ 6 \\ 2 \end{array} \right)$$ where \(\mu\) is a scalar parameter. The plane \(\Pi _ { 2 }\) contains the line \(l _ { 1 }\) and the line \(l _ { 2 }\)
3. Determine a vector equation for the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) The plane \(\Pi _ { 3 }\) has equation r. \(\left( \begin{array} { l } 1 \\ 1 \\ a \end{array} \right) = b\) where \(a\) and \(b\) are constants.
   Given that the planes \(\Pi _ { 1 } , \Pi _ { 2 }\) and \(\Pi _ { 3 }\) form a sheaf,
4. determine the value of \(a\) and the value of \(b\).

1. 1. The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ - 3 \\ 1 \end{array} \right) + \lambda \left( \begin{array} { r } 3 \\ 4 \\ - 1 \end{array} \right)\)

The line \(l _ { 2 }\) has equation \(\mathbf { r } = \left( \begin{array} { c } 13 \\ 5 \\ 8 \end{array} \right) + \mu \left( \begin{array} { r } 1 \\ - 2 \\ 5 \end{array} \right)\) where \(\lambda\) and \(\mu\) are scalar parameters.
The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(P\).

1. Determine the coordinates of \(P\). Given that the plane \(\Pi\) contains both \(l _ { 1 }\) and \(l _ { 2 }\)
2. determine a Cartesian equation for \(\Pi\).
   (ii) Determine a Cartesian equation for each of the two lines that
   • pass through \(( 0,0,0 )\)
   • make an angle of \(60 ^ { \circ }\) with the \(x\)-axis
   • make an angle of \(45 ^ { \circ }\) with the \(y\)-axis

2. The general solution of the first order recurrence relation $$u _ { n + 1 } + a u _ { n } = b n ^ { 2 } + c n + d \quad n \geqslant 0$$ is given by $$u _ { n } = A ( 3 ) ^ { n } + 5 n ^ { 2 } + 1$$ where \(A\) is an arbitrary non-zero constant.
By considering expressions for \(u _ { n + 1 }\) and \(u _ { n }\), find the values of the constants \(a , b , c\) and \(d\).

8. A sequence \(\left\{ u _ { n } \right\}\), where \(n \geqslant 0\), satisfies the recurrence relation $$2 u _ { n + 2 } + 5 u _ { n + 1 } = 3 u _ { n } + 8 n + 2$$

1. Find the general solution of this recurrence relation.
   (5) A particular solution of this recurrence relation has \(u _ { 0 } = 1\) and \(u _ { 1 } = k\), where \(k\) is a positive constant. All terms of the sequence are positive.
2. Determine the value of \(k\).
   (3)

1. (a) Find the general solution of the recurrence relation

$$u _ { n + 2 } = u _ { n + 1 } + u _ { n } , \quad n \geqslant 1$$ Given that \(u _ { 1 } = 1\) and \(u _ { 2 } = 1\) (b) find the particular solution of the recurrence relation.

4 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\frac { x - 7 } { 2 } = \frac { y - 1 } { - 1 } = \frac { z - 6 } { 3 }\) and \(\frac { x - 2 } { 1 } = \frac { y - 6 } { 2 } = \frac { z + 2 } { 1 }\) respectively.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
2. Find the cartesian equation of the plane that contains \(l _ { 1 }\) and \(l _ { 2 }\).

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

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

6 The equation of a plane \(\Pi\) is \(x - 2 y - z = 30\).

1. Find the acute angle between the line \(\mathbf { r } = \left( \begin{array} { c } 3 \\ 2 \\ - 5 \end{array} \right) + \lambda \left( \begin{array} { r } - 5 \\ 3 \\ 2 \end{array} \right)\) and \(\Pi\).
2. Determine the geometrical relationship between the line \(\mathbf { r } = \left( \begin{array} { l } 1 \\ 4 \\ 2 \end{array} \right) + \mu \left( \begin{array} { r } 3 \\ - 1 \\ 5 \end{array} \right)\) and \(\Pi\).

7. The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\mathbf { r } = \left( \begin{array} { r } 1 \\ - 1 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ 3 \\ 4 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } \alpha \\ - 4 \\ 0 \end{array} \right) + \mu \left( \begin{array} { l } 0 \\ 3 \\ 2 \end{array} \right) .$$ If the lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect, find

1. the value of \(\alpha\),
2. an equation for the plane containing the lines \(l _ { 1 }\) and \(l _ { 2 }\), giving your answer in the form \(a x + b y + c z + d = 0\), where \(a , b , c\) and \(d\) are constants. For other values of \(\alpha\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) do not intersect and are skew lines.
   Given that \(\alpha = 2\),
3. find the shortest distance between the lines \(l _ { 1 }\) and \(l _ { 2 }\).

7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = \left[ \begin{array} { r } 8 \\ 6 \\ - 9 \end{array} \right] + \lambda \left[ \begin{array} { r } 3 \\ - 3 \\ - 1 \end{array} \right]\) and \(\mathbf { r } = \left[ \begin{array} { r } - 4 \\ 0 \\ 11 \end{array} \right] + \mu \left[ \begin{array} { r } 1 \\ 2 \\ - 3 \end{array} \right]\) respectively.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular.
2. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the coordinates of the point of intersection, \(P\).
3. The point \(A ( - 4,0,11 )\) lies on \(l _ { 2 }\). The point \(B\) on \(l _ { 1 }\) is such that \(A P = B P\). Find the length of \(A B\).

3 The line \(L _ { 1 }\) passes through the points \(( 2 , - 3,1 )\) and \(( - 1 , - 2 , - 4 )\). The line \(L _ { 2 }\) passes through the point \(( 3,2 , - 9 )\) and is parallel to the vector \(4 \mathbf { i } - 4 \mathbf { j } + 5 \mathbf { k }\).

1. Find an equation for \(L _ { 1 }\) in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\).
2. Prove that \(L _ { 1 }\) and \(L _ { 2 }\) are skew.

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
      1. (ii) Find the position vector of the point of intersection.
         15
   2. 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
   3. 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 \(l _ { 2 }\)

7 A lighting engineer is setting up part of a display inside a large building. The diagram shows a plan view of the area in which he is working. He has two lights, which project narrow beams of light. One is set up at a point 3 metres above the point \(A\) and the beam from this light hits the wall 23 metres above the point \(D\). The other is set up 1 metre above the point \(B\) and the beam from this light hits the wall 29 metres above the point \(C\). \includegraphics[max width=\textwidth, alt={}, center]{e61d0202-49c9-4ed9-9fa3-f10734e17463-10_776_1301_826_392} 7

1. By creating a suitable model, show that the beams of light intersect. 7
2. Find the angle between the two beams of light.
   [0pt] [3 marks]
   7
3. State one way in which the model you created in part (a) could be refined.
   [0pt] [1 mark]

11 The lines \(l _ { 1 } , l _ { 2 }\) and \(l _ { 3 }\) are defined as follows. $$\begin{aligned} & l _ { 1 } : \left( \mathbf { r } - \left[ \begin{array} { c } 1 \\ 5 \\ - 1 \end{array} \right] \right) \times \left[ \begin{array} { c } - 2 \\ 1 \\ - 3 \end{array} \right] = \mathbf { 0 } \\ & l _ { 2 } : \left( \mathbf { r } - \left[ \begin{array} { c } - 3 \\ 2 \\ 7 \end{array} \right] \right) \times \left[ \begin{array} { c } 2 \\ - 1 \\ 3 \end{array} \right] = \mathbf { 0 } \\ & l _ { 3 } : \left( \mathbf { r } - \left[ \begin{array} { c } - 5 \\ 12 \\ - 4 \end{array} \right] \right) \times \left[ \begin{array} { l } 4 \\ 0 \\ 9 \end{array} \right] = \mathbf { 0 } \end{aligned}$$ 11

1. 1. Explain how you know that two of the lines are parallel.
      

      11 (ii) Show that the perpendicular distance between these two parallel lines is 7.95 units, correct to three significant figures. [5 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

      11
2. Show that the lines \(l _ { 1 }\) and \(l _ { 3 }\) meet, and find the coordinates of their point of intersection. \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-23_2488_1716_219_153}

8. The owner of a new company models the number of customers that the company will have at the end of each month. The owner assumes that

• a constant proportion, \(p\) (where \(0 < p < 1\) ), of the previous month's customers will be retained for the next month
• a constant number of new customers, \(k\), will be added each month.

Let \(u _ { n }\) (where \(n \geqslant 1\) ) represent the number of customers that the company will have at the end of \(n\) months. The company has 5000 customers at the end of the first month.

1. By setting up a first order recurrence relation for \(u _ { n + 1 }\) in terms of \(u _ { n }\), determine an expression for \(u _ { n }\) in terms of \(n , p\) and \(k\). The owner believes that \(95 \%\) of the previous month's customers will be retained each month and that there will be 10000 new customers each month. According to the model, the company will first have at least 135000 customers by the end of the \(m\) th month.
2. Using logarithms, determine the value of \(m\). Please check the examination details below before entering your candidate information \section*{Further Mathematics} Advanced
   PAPER 4D: Decision Mathematics 2 \section*{Answer Book} Do not return the question paper with the answer book.
   1.

   	1	2	3	4
   A	32	45	34	48
   B	37	39	50	46
   C	46	44	40	42
   D	43	45	48	52

   You may not need to use all of these tables.

   	1	2	3	4
   A
   B
   C
   D

   	1	2	3	4
   A
   B
   C
   D

   	1	2	3	4
   A
   B
   C
   D

   	1	2	3	4
   A
   B
   C
   D

   VALV SIHI NI IIIIIM ION OC
   VIAV SIUL NI JAIIM ION OC
   VIAV SIHI NI III IM ION OC

   	1	2	3	4
   A
   B
   C
   D

   	1	2	3	4
   \(A\)
   \(B\)
   \(C\)
   \(D\)

   	1	2	3	4
   A
   B
   C
   D

   	1	2	3	4
   A
   B
   C
   D

   	1	2	3	4
   A
   B
   C
   D

   2.

   VAMV SIHI NI IIIHM ION OO

   VIAV SIHI NI JIIIM I ON OC

   VJYV SIHI NI JIIIM ION OC

   3. 4. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cea07472-f93b-4a7b-b362-89fb8c0af4a9-16_936_1317_255_376} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} 5.

3. 	\(R\)	\(S\)	\(T\)
   \(A\)
   \(B\)
   \(C\)
   \(D\)

   	\(R\)	\(S\)	\(T\)
   \(A\)
   \(B\)
   \(C\)
   \(D\)

   	\(R\)	\(S\)	\(T\)
   \(A\)
   \(B\)
   \(C\)
   \(D\)

   	\(R\)	\(S\)	\(T\)
   \(A\)
   \(B\)
   \(C\)
   \(D\)

   	\(R\)	\(S\)	\(T\)
   \(A\)
   \(B\)
   \(C\)
   \(D\)

   6.

   Stage	State	Action	Destination	Value
   May	2	0	0	160
   	1	1	0	80 + 35
   	0	2	0	70

   Stage	State	Action	Destination	Value

   Month	January	February	March	April	May
   Number made

   Minimum cost: \(\_\_\_\_\) 7. Player A \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Player B}

   	Option W	Option X	Option Y	Option Z
   Option Q	4	3	-1	-2
   Option R	-3	5	-4	\(k\)
   Option S	-1	6	3	-3

   \end{table} The tableau for (d)(ii) can be found at the bottom of page 16

   b.v.				□	□	□	□		Value
   \includegraphics[max width=\textwidth, alt={}]{cea07472-f93b-4a7b-b362-89fb8c0af4a9-24_56_77_2348_182}		□	□
   P

   8.

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 \(l _ { 2 }\).

8 The plane \(\Pi _ { 1 }\) has equation $$\mathbf { r } = \left( \begin{array} { l } 5 \\ 1 \\ 0 \end{array} \right) + s \left( \begin{array} { r } - 4 \\ 1 \\ 3 \end{array} \right) + t \left( \begin{array} { l } 0 \\ 1 \\ 2 \end{array} \right)$$

1. Find a cartesian equation of \(\Pi _ { 1 }\).
   The plane \(\Pi _ { 2 }\) has equation \(3 x + y - z = 3\).
2. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in degrees.
3. Find an equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\). [5]