Parallel and perpendicular lines

8 The points \(A , B , C\) and \(D\) have position vectors \(3 \mathbf { i } + 2 \mathbf { k } , 2 \mathbf { i } - 2 \mathbf { j } + 5 \mathbf { k } , 2 \mathbf { j } + 7 \mathbf { k }\) and \(- 2 \mathbf { i } + 10 \mathbf { j } + 7 \mathbf { k }\) respectively.

1. Use a scalar product to show that \(B A\) and \(B C\) are perpendicular.
2. Show that \(B C\) and \(A D\) are parallel and find the ratio of the length of \(B C\) to the length of \(A D\).

7 \includegraphics[max width=\textwidth, alt={}, center]{1a4ddaa9-1ec2-4138-bfcb-a482fe6c942f-3_394_750_260_699} The diagram shows a trapezium \(A B C D\) in which \(B A\) is parallel to \(C D\). The position vectors of \(A , B\) and \(C\) relative to an origin \(O\) are given by $$\overrightarrow { O A } = \left( \begin{array} { l } 3 \\ 4 \\ 0 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { l } 1 \\ 3 \\ 2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 4 \\ 5 \\ 6 \end{array} \right)$$

1. Use a scalar product to show that \(A B\) is perpendicular to \(B C\).
2. Given that the length of \(C D\) is 12 units, find the position vector of \(D\).

14
- 6
- 13 \end{array} \right) + \lambda \left( \begin{array} { r } - 2
1
4 \end{array} \right) \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } p
- 7
4 \end{array} \right) + \mu \left( \begin{array} { l } q
2
1 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(p\) and \(q\) are constants. Given that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular,

1. show that \(q = 3\) Given further that \(l _ { 1 }\) and \(l _ { 2 }\) intersect at point \(X\), find
2. the value of \(p\),
3. the coordinates of \(X\). The point \(A\) lies on \(l _ { 1 }\) and has position vector \(\left( \begin{array} { r } 6 \\ - 2 \\ 3 \end{array} \right)\)\\ Given that point \(B\) also lies on \(l _ { 1 }\) and that \(A B = 2 A X\)
4. find the two possible position vectors of \(B\).\\ 12. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{03548211-79cb-4629-b6ca-aa9dfcc77a33-21_615_732_233_605} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation $$y = \frac { x ^ { 2 } \ln x } { 3 } - 2 x + 4 , \quad x > 0$$ The finite region \(S\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(x\)-axis and the lines with equations \(x = 1\) and \(x = 3\)
5. Complete the table below with the value of \(y\) corresponding to \(x = 2\). Give your answer to 4 decimal places.

   \(x\)	1	1.5	2	2.5	3
   \(y\)	2	1.3041		0.9089	1.2958

6. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate for the area of \(S\), giving your answer to 3 decimal places.
7. Use calculus to find the exact area of \(S\). Give your answer in the form \(\frac { a } { b } + \ln c\), where \(a , b\) and \(c\) are integers.
8. Hence calculate the percentage error in using your answer to part (b) to estimate the area of \(S\). Give your answer to one decimal place.
9. Explain how the trapezium rule could be used to obtain a more accurate estimate for the area of \(S\). 13. (a) Express \(10 \cos \theta - 3 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\) Give the exact value of \(R\) and give the value of \(\alpha\) to 2 decimal places. Alana models the height above the ground of a passenger on a Ferris wheel by the equation $$H = 12 - 10 \cos ( 30 t ) ^ { \circ } + 3 \sin ( 30 t ) ^ { \circ }$$ where the height of the passenger above the ground is \(H\) metres at time \(t\) minutes after the wheel starts turning. \includegraphics[max width=\textwidth, alt={}, center]{03548211-79cb-4629-b6ca-aa9dfcc77a33-23_419_567_516_1160}
10. Calculate
    1. the maximum value of \(H\) predicted by this model,
    2. the value of \(t\) when this maximum first occurs. Give each answer to 2 decimal places.
11. Calculate the value of \(t\) when the passenger is 18 m above the ground for the first time. Give your answer to 2 decimal places.
12. Determine the time taken for the Ferris wheel to complete two revolutions.

4. With respect to a fixed origin \(O\) the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$l _ { 1 } : \quad \mathbf { r } = \left( \begin{array} { c }

9 Lines \(L _ { 1 } , L _ { 2 }\) and \(L _ { 3 }\) have vector equations $$\begin{aligned} & L _ { 1 } : \mathbf { r } = ( 5 \mathbf { i } - \mathbf { j } - 2 \mathbf { k } ) + s ( - 6 \mathbf { i } + 8 \mathbf { j } - 2 \mathbf { k } ) , \\ & L _ { 2 } : \mathbf { r } = ( 3 \mathbf { i } - 8 \mathbf { j } ) + t ( \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } ) , \\ & L _ { 3 } : \mathbf { r } = ( 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) + u ( 3 \mathbf { i } + c \mathbf { j } + \mathbf { k } ) . \end{aligned}$$

1. Calculate the acute angle between \(L _ { 1 }\) and \(L _ { 2 }\).
2. Given that \(L _ { 1 }\) and \(L _ { 3 }\) are parallel, find the value of \(c\).
3. Given instead that \(L _ { 2 }\) and \(L _ { 3 }\) intersect, find the value of \(c\). 4

5 The equations of three lines are as follows. $$\begin{array} { l l } \text { Line } A : & \mathbf { r } = \mathbf { i } + 4 \mathbf { j } + \mathbf { k } + s ( - \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } ) \\ \text { Line } B : & \mathbf { r } = 2 \mathbf { i } + 8 \mathbf { j } + 2 \mathbf { k } + t ( \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } ) \\ \text { Line } C : & \mathbf { r } = - \mathbf { i } + 19 \mathbf { j } + 15 \mathbf { k } + u ( 2 \mathbf { i } - 4 \mathbf { j } - 4 \mathbf { k } ) \end{array}$$

1. Show that lines \(A\) and \(B\) are skew.
2. Determine, giving reasons, the geometrical relationship between lines \(A\) and \(C\).

3. Relative to a fixed origin,

• point \(A\) has position vector \(- 2 \mathbf { i } + 4 \mathbf { j } + 7 \mathbf { k }\)
• point \(B\) has position vector \(- \mathbf { i } + 3 \mathbf { j } + 8 \mathbf { k }\)
• point \(C\) has position vector \(\mathbf { i } + \mathbf { j } + 4 \mathbf { k }\)
• point \(D\) has position vector \(- \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }\) a. Show that \(\overrightarrow { A B }\) and \(\overrightarrow { C D }\) are parallel and the ratio \(\overrightarrow { A B } : \overrightarrow { C D }\) in its simplest form.
  b. Hence describe the quadrilateral \(A B C D\).

1. Relative to a fixed origin \(O\)

• point \(A\) has position vector \(2 \mathbf { i } + 5 \mathbf { j } - 6 \mathbf { k }\)
• point \(B\) has position vector \(3 \mathbf { i } - 3 \mathbf { j } - 4 \mathbf { k }\)
• point \(C\) has position vector \(2 \mathbf { i } - 16 \mathbf { j } + 4 \mathbf { k }\)
  1. Find \(\overrightarrow { A B }\)
  2. Show that quadrilateral \(O A B C\) is a trapezium, giving reasons for your answer.

7. Two submarines are travelling in straight lines through the ocean. Relative to a fixed origin, the vector equations of the two lines, \(l _ { 1 }\) and \(l _ { 2 }\), along which they travel are $$\text { and } \quad \begin{aligned} & \mathbf { r } = 3 \mathbf { i } + 4 \mathbf { j } - 5 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } ) \\ & \mathbf { r } = 9 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } + \mu ( 4 \mathbf { i } + \mathbf { j } - \mathbf { k } ) , \end{aligned}$$ where \(\lambda\) and \(\mu\) are scalars.

1. Show that the submarines are moving in perpendicular directions.
2. Given that \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(A\), find the position vector of \(A\). The point \(b\) has position vector \(10 \mathbf { j } - 11 \mathbf { k }\).
3. Show that only one of the submarines passes through the point \(B\).
4. Given that 1 unit on each coordinate axis represents 100 m , find, in km , the distance \(A B\).

4. Relative to a fixed origin, two lines have the equations $$\begin{aligned} & \mathbf { r } = \left( \begin{array} { c } 7
0
- 3 \end{array} \right) + \lambda \left( \begin{array} { c }

2. The position vectors of the points \(A , B , C , D\) are given by \(\mathbf { a } = 2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k }\), \(\mathbf { b } = 4 \mathbf { j } + 5 \mathbf { k }\), \(\mathbf { c } = 7 \mathbf { i } - 3 \mathbf { k }\), \(\mathbf { d } = - 3 \mathbf { i } - \mathbf { j } - 5 \mathbf { k }\),
respectively.

1. Find the vector equations of the lines \(A B\) and \(C D\).
2. Determine whether or not the lines \(A B\) and \(C D\) are perpendicular.

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)

6. Two submarines are travelling in straight lines through the ocean. Relative to a fixed origin, the vector equations of the two lines, \(l _ { 1 }\) and \(l _ { 2 }\), along which they travel are $$\begin{aligned} \mathbf { r } & = 3 \mathbf { i } + 4 \mathbf { j } - 5 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } ) \\ \text { and } \mathbf { r } & = 9 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } + \mu ( 4 \mathbf { i } + \mathbf { j } - \mathbf { k } ) , \end{aligned}$$ where \(\lambda\) and \(\mu\) are scalars.

1. Show that the submarines are moving in perpendicular directions.
2. Given that \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(A\), find the position vector of \(A\). The point \(b\) has position vector \(10 \mathbf { j } - 11 \mathbf { k }\).
3. Show that only one of the submarines passes through the point \(B\).
4. Given that 1 unit on each coordinate axis represents 100 m , find, in km , the distance \(A B\).

7 The quadrilateral \(A B C D\) has vertices \(A ( 2,1,3 ) , B ( 6,5,3 ) , C ( 6,1 , - 1 )\) and \(D ( 2 , - 3 , - 1 )\).
The line \(l _ { 1 }\) has vector equation \(\mathbf { r } = \left[ \begin{array} { r } 6 \\ 1 \\ - 1 \end{array} \right] + \lambda \left[ \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right]\).

1. 1. Find the vector \(\overrightarrow { A B }\).
   2. Show that the line \(A B\) is parallel to \(l _ { 1 }\).
   3. Verify that \(D\) lies on \(l _ { 1 }\).
2. The line \(l _ { 2 }\) passes through \(D ( 2 , - 3 , - 1 )\) and \(M ( 4,1,1 )\).
   1. Find the vector equation of \(l _ { 2 }\).
   2. Find the angle between \(l _ { 2 }\) and \(A C\).

13 Line \(l _ { 1 }\) has equation $$\frac { x - 2 } { 3 } = \frac { 1 - 2 y } { 4 } = - z$$ and line \(l _ { 2 }\) has equation $$\mathbf { r } = \left[ \begin{array} { c } - 7 \\ 4 \\ - 2 \end{array} \right] + \mu \left[ \begin{array} { c } 12 \\ a + 3 \\ 2 b \end{array} \right]$$ 13

1. In the case when \(l _ { 1 }\) and \(l _ { 2 }\) are parallel, show that \(a = - 11\) and find the value of \(b\).
   
   \includegraphics[max width=\textwidth, alt={}]{86aa9e6f-261c-40d4-8271-a0dc560d8a72-19_2484_1712_219_150}

3 The line \(L\) has equation \(\mathbf { r } = \left[ \begin{array} { l } 3 \\ 2 \\ 0 \end{array} \right] + \lambda \left[ \begin{array} { c } - 1 \\ - 2 \\ 5 \end{array} \right]\) Which of the following lines is perpendicular to the line \(L\) ?
Tick \(( \checkmark )\) one box. $$\begin{aligned} & \mathbf { r } = \left[ \begin{array} { c } 2 \\ - 3 \\ 4 \end{array} \right] + \mu \left[ \begin{array} { c } 1 \\ 2 \\ - 5 \end{array} \right] \\ & \mathbf { r } = \left[ \begin{array} { l } 1 \\ 0 \\ 1 \end{array} \right] + \mu \left[ \begin{array} { c } 2 \\ - 3 \\ 1 \end{array} \right] \\ & \mathbf { r } = \left[ \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right] + \mu \left[ \begin{array} { l } 1 \\ 1 \\ 2 \end{array} \right] \\ & \mathbf { r } = \left[ \begin{array} { l } 0 \\ 3 \\ 2 \end{array} \right] + \mu \left[ \begin{array} { l } 4 \\ 3 \\ 2 \end{array} \right] \end{aligned}$$ □
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