Vectors 3D & Lines

5 Two vectors, \(\mathbf { u }\) and \(\mathbf { v }\), are such that $$\mathbf { u } = \left( \begin{array} { l } q
2

2 The two vectors \(\mathbf { a }\) and \(\mathbf { b }\) are such that \(\mathbf { a } \cdot \mathbf { b } = 0\) State the angle between the vectors \(\mathbf { a }\) and \(\mathbf { b }\) Circle your answer.
[0pt] [1 mark] \(0 ^ { \circ } 45 ^ { \circ } 90 ^ { \circ } 180 ^ { \circ }\)

2 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r }

1
6 \end{array} \right) + \lambda \left( \begin{array} { r }

1. Relative to a fixed origin \(O\), the point \(A\) has position vector \(\mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k }\) and the point \(B\) has position vector \(- 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }\). The points \(A\) and \(B\) lie on a straight line \(l\).
   1. Find \(\overrightarrow { A B }\).
   2. Find a vector equation of \(l\).
   The point \(C\) has position vector \(2 \mathbf { i } + p \mathbf { j } - 4 \mathbf { k }\) with respect to \(O\), where \(p\) is a constant. Given that \(A C\) is perpendicular to \(l\), find
2. the value of \(p\),
3. the distance \(A C\).

The line \(l _ { 1 }\) passes through the point \(A\) whose position vector is \(4 \mathbf { i } + 7 \mathbf { j } - \mathbf { k }\) and is parallel to the vector \(3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }\). The line \(l _ { 2 }\) passes through the point \(B\) whose position vector is \(\mathbf { i } + 7 \mathbf { j } + 11 \mathbf { k }\) and is parallel to the vector \(\mathbf { i } - 6 \mathbf { j } - 2 \mathbf { k }\). The points \(P\) on \(l _ { 1 }\) and \(Q\) on \(l _ { 2 }\) are such that \(P Q\) is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\). Find the position vectors of \(P\) and \(Q\). Find the shortest distance between the line through \(A\) and \(B\) and the line through \(P\) and \(Q\), giving your answer correct to 3 significant figures.

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

5 Verify that the point \(( - 1,6,5 )\) lies on both the lines $$\mathbf { r } = \left( \begin{array} { r } 1
2
- 1 \end{array} \right) + \lambda \left( \begin{array} { r } - 1
2
3 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { l } 0

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

5 The points A , B and C have coordinates \(( 2,0 , - 1 ) , ( 4,3 , - 6 )\) and \(( 9,3 , - 4 )\) respectively.

1. Show that AB is perpendicular to BC .
2. Find the area of triangle ABC .

6 Relative to an origin \(O\), the position vector of \(A\) is \(3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }\) and the position vector of \(B\) is \(7 \mathbf { i } - 3 \mathbf { j } + \mathbf { k }\).

1. Show that angle \(O A B\) is a right angle.
2. Find the area of triangle \(O A B\).

7
0
7 \end{array} \right) + \mu \left( \begin{array} { r } 3
- 5
4 \end{array} \right) , \quad \text { where } \mu \text { is a scalar parameter. }$$

1. Find the value of the constant \(p\).
2. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the position vector of their point of intersection, \(C\).
3. Find the size of the angle \(A C B\), giving your answer in degrees to 3 significant figures.
4. Find the area of the triangle \(A B C\), giving your answer to 3 significant figures.\\ 7. The rate of increase of the number, \(N\), of fish in a lake is modelled by the differential equation $$\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { ( k t - 1 ) ( 5000 - N ) } { t } \quad t > 0 , \quad 0 < N < 5000$$ In the given equation, the time \(t\) is measured in years from the start of January 2000 and \(k\) is a positive constant.
5. By solving the differential equation, show that $$N = 5000 - A t \mathrm { e } ^ { - k t }$$ where \(A\) is a positive constant. After one year, at the start of January 2001, there are 1200 fish in the lake. After two years, at the start of January 2002, there are 1800 fish in the lake.
6. Find the exact value of the constant \(A\) and the exact value of the constant \(k\).
7. Hence find the number of fish in the lake after five years. Give your answer to the nearest hundred fish.

1. With respect to a fixed origin \(O\), the line \(l\) has equation

$$\mathbf { r } = \left( \begin{array} { c }

9 \includegraphics[max width=\textwidth, alt={}, center]{9f17f7b8-b54d-467d-be26-21c599ce6ca2-4_724_1488_257_330} The diagram shows a cuboid \(O A B C D E F G\) with a horizontal base \(O A B C\) in which \(O A = 4 \mathrm {~cm}\) and \(A B = 15 \mathrm {~cm}\). The height \(O D\) of the cuboid is 2 cm . The point \(X\) on \(A B\) is such that \(A X = 5 \mathrm {~cm}\) and the point \(P\) on \(D G\) is such that \(D P = p \mathrm {~cm}\), where \(p\) is a constant. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O D\) respectively.

1. Find the possible values of \(p\) such that angle \(O P X = 90 ^ { \circ }\).
2. For the case where \(p = 9\), find the unit vector in the direction of \(\overrightarrow { X P }\).
3. A point \(Q\) lies on the face \(C B F G\) and is such that \(X Q\) is parallel to \(A G\). Find \(\overrightarrow { X Q }\).

6 The points \(A\) and \(B\) have coordinates \(( 3,2,10 )\) and \(( 5 , - 2,4 )\) respectively.
The line \(l\) passes through \(A\) and has equation \(\mathbf { r } = \left[ \begin{array} { r } 3 \\ 2 \\ 10 \end{array} \right] + \lambda \left[ \begin{array} { r } 3 \\ 1 \\ - 2 \end{array} \right]\).

1. Find the acute angle between \(l\) and the line \(A B\).
2. The point \(C\) lies on \(l\) such that angle \(A B C\) is \(90 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{fdd3905e-11f7-4b20-adfe-4c686018a221-12_360_339_762_852} Find the coordinates of \(C\).
3. The point \(D\) is such that \(B D\) is parallel to \(A C\) and angle \(B C D\) is \(90 ^ { \circ }\). The point \(E\) lies on the line through \(B\) and \(D\) and is such that the length of \(D E\) is half that of \(A C\). Find the coordinates of the two possible positions of \(E\).
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6 \end{array} \right) \quad \text { and } \quad \mathbf { v } = \left( \begin{array} { c }

6 \end{array} \right) \quad \text { and } \quad \mathbf { v } = \left( \begin{array} { c }

14. The three dimensional non-zero vector \(\boldsymbol { u }\) has the following properties:

• The angle \(\theta\) between \(\boldsymbol { u }\) and the vector \(\left( \begin{array} { l } 1 \\ 5 \\ 9 \end{array} \right)\) is acute.
• The (non-reflex) angle between \(\boldsymbol { u }\) and the vector \(\left( \begin{array} { l } 9 \\ 5 \\ 1 \end{array} \right)\) is \(2 \theta\).
• \(\boldsymbol { u }\) is perpendicular to the vector \(\left( \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right)\).

Find the angle \(\theta\).
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8 \includegraphics[max width=\textwidth, alt={}, center]{d178603a-f59a-4986-b5ab-b47eceedb2fc-12_595_748_260_699} The diagram shows a solid figure \(O A B C D E F\) having a horizontal rectangular base \(O A B C\) with \(O A = 6\) units and \(A B = 3\) units. The vertical edges \(O F , A D\) and \(B E\) have lengths 6 units, 4 units and 4 units respectively. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O F\) respectively.

1. Find \(\overrightarrow { D F }\).
2. Find the unit vector in the direction of \(\overrightarrow { E F }\).
3. Use a scalar product to find angle \(E F D\).

5 \includegraphics[max width=\textwidth, alt={}, center]{b2cefbd6-6e89-495a-9f42-60f76c8c5975-3_1070_754_255_699} The diagram shows a solid cylinder standing on a horizontal circular base, centre \(O\) and radius 4 units. The line \(B A\) is a diameter and the radius \(O C\) is at \(90 ^ { \circ }\) to \(O A\). Points \(O ^ { \prime } , A ^ { \prime } , B ^ { \prime }\) and \(C ^ { \prime }\) lie on the upper surface of the cylinder such that \(O O ^ { \prime } , A A ^ { \prime } , B B ^ { \prime }\) and \(C C ^ { \prime }\) are all vertical and of length 12 units. The mid-point of \(B B ^ { \prime }\) is \(M\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O O ^ { \prime }\) respectively.

1. Express each of the vectors \(\overrightarrow { M O }\) and \(\overrightarrow { M C ^ { \prime } }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Hence find the angle \(O M C ^ { \prime }\).

9 \includegraphics[max width=\textwidth, alt={}, center]{ae57d8f1-5a0d-426c-952d-e8b99c6aeaba-4_582_1072_255_541} The diagram shows a pyramid \(O A B C P\) in which the horizontal base \(O A B C\) is a square of side 10 cm and the vertex \(P\) is 10 cm vertically above \(O\). The points \(D , E , F , G\) lie on \(O P , A P , B P , C P\) respectively and \(D E F G\) is a horizontal square of side 6 cm . The height of \(D E F G\) above the base is \(a \mathrm {~cm}\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O D\) respectively.

1. Show that \(a = 4\).
2. Express the vector \(\overrightarrow { B G }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
3. Use a scalar product to find angle \(G B A\).

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 }

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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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 \(D\).

8 The points \(A ( 3,2,1 ) , B ( 5,4 , - 3 ) , C ( 3,17 , - 4 )\) and \(D ( 1,6,3 )\) form a quadrilateral \(A B C D\).

1. Show that \(A B = A D\).
2. Find a vector equation of the line through \(A\) and the mid-point of \(B D\).
3. Show that \(C\) lies on the line found in part (ii).
4. What type of quadrilateral is \(A B C D\) ?

1. Relative to a fixed origin \(O\),
   the point \(A\) has position vector \(\mathbf { i } + 7 \mathbf { j } - 2 \mathbf { k }\),
   the point \(B\) has position vector \(4 \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k }\),
   and the point \(C\) has position vector \(2 \mathbf { i } + 10 \mathbf { j } + 9 \mathbf { k }\).
   Given that \(A B C D\) is a parallelogram,
   1. find the position vector of point \(D\).
   The vector \(\overrightarrow { A X }\) has the same direction as \(\overrightarrow { A B }\).
   Given that \(| \overrightarrow { A X } | = 10 \sqrt { 2 }\),
2. find the position vector of \(X\).

6 The line \(l _ { 1 }\) passes through the point \(A ( 0,6,9 )\) and the point \(B ( 4 , - 6 , - 11 )\).
The line \(l _ { 2 }\) has equation \(\mathbf { r } = \left[ \begin{array} { r } - 1 \\ 5 \\ - 2 \end{array} \right] + \lambda \left[ \begin{array} { r } 3 \\ - 5 \\ 1 \end{array} \right]\).

1. The acute angle between the lines \(l _ { 1 }\) and \(l _ { 2 }\) is \(\theta\). Find the value of \(\cos \theta\) as a fraction in its lowest terms.
2. Show that the lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the coordinates of the point of intersection.
3. The points \(C\) and \(D\) lie on line \(l _ { 2 }\) such that \(A C B D\) is a parallelogram. \includegraphics[max width=\textwidth, alt={}, center]{c42685e9-bfa4-48d4-8abb-13e88a4b765e-12_392_949_1018_548} The length of \(A B\) is three times the length of \(C D\).
   Find the coordinates of the points \(C\) and \(D\).
   [0pt] [5 marks] \(7 \quad\) A curve \(C\) is defined by the parametric equations $$x = \frac { 4 - \mathrm { e } ^ { 2 - 6 t } } { 4 } , \quad y = \frac { \mathrm { e } ^ { 3 t } } { 3 t } , \quad t \neq 0$$

4. With respect to a fixed origin \(O\), the line \(l _ { 1 }\) has vector equation $$\mathbf { r } = \left( \begin{array} { r } - 9

4. With respect to a fixed origin \(O\), the line \(l _ { 1 }\) has vector equation $$\mathbf { r } = \left( \begin{array} { r } - 9

7. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = ( 13 \mathbf { i } + 15 \mathbf { j } - 8 \mathbf { k } ) + \lambda ( 3 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k } ) \\ & l _ { 2 } : \mathbf { r } = ( 7 \mathbf { i } - 6 \mathbf { j } + 14 \mathbf { k } ) + \mu ( 2 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k } ) \end{aligned}$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) meet and find the position vector of their point of intersection, \(B\).
2. Find the acute angle between the lines \(l _ { 1 }\) and \(l _ { 2 }\) The point \(A\) has position vector \(- 5 \mathbf { i } - 3 \mathbf { j } + 16 \mathbf { k }\)
3. Show that \(A\) lies on \(l _ { 1 }\) The point \(C\) lies on the line \(l _ { 1 }\) where \(\overrightarrow { A B } = \overrightarrow { B C }\)
4. Find the position vector of \(C\).
   
   \section*{"}

3 The points \(A\) and \(B\) have position vectors \(\mathbf { a } = \left( \begin{array} { r } 3 \\ 2 \\ - 1 \end{array} \right)\) and \(\mathbf { b } = \left( \begin{array} { r } - 1 \\ 4 \\ 8 \end{array} \right)\) respectively.
Show that the exact value of the distance \(A B\) is \(\sqrt { \mathbf { 1 0 1 } }\).

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

• the point \(A\) has position vector \(5 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }\)
• the point \(B\) has position vector \(2 \mathbf { i } + 4 \mathbf { j } + a \mathbf { k }\) where \(a\) is a positive integer.
  1. Show that \(| \overrightarrow { O A } | = \sqrt { 38 }\)
  2. Find the smallest value of \(a\) for which

$$| \overrightarrow { O B } | > | \overrightarrow { O A } |$$

9 Relative to an origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 8 \\ - 6 \\ 5 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } - 10 \\ 3 \\ - 13 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 2 \\ - 3 \\ - 1 \end{array} \right)$$ A fourth point, \(D\), is such that the magnitudes \(| \overrightarrow { A B } | , | \overrightarrow { B C } |\) and \(| \overrightarrow { C D } |\) are the first, second and third terms respectively of a geometric progression.

1. Find the magnitudes \(| \overrightarrow { A B } | , | \overrightarrow { B C } |\) and \(| \overrightarrow { C D } |\).
2. Given that \(D\) is a point lying on the line through \(B\) and \(C\), find the two possible position vectors of the point \(D\).

7 A particle \(P\) moves with constant acceleration \(( 3 \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\). At time \(t = 0\) seconds \(P\) is at the origin. At time \(t = 4\) seconds \(P\) has velocity \(( 2 \mathbf { i } + 4 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).

1. Find the displacement vector of \(P\) at time \(t = 4\) seconds.
2. Find the speed of \(P\) at time \(t = 0\) seconds.

7 A particle \(P\) moves with constant acceleration \(( 3 \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\). At time \(t = 0\) seconds \(P\) is at the origin. At time \(t = 4\) seconds \(P\) has velocity \(( 2 \mathbf { i } + 4 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).

1. Find the displacement vector of \(P\) at time \(t = 4\) seconds.
2. Find the speed of \(P\) at time \(t = 0\) seconds.

1. An aeroplane leaves a runway and moves with a constant speed of \(V \mathrm {~km} / \mathrm { h }\) due north along a straight path inclined at an angle \(\arctan \left( \frac { 3 } { 4 } \right)\) to the horizontal.

A light aircraft is moving due north in a straight horizontal line in the same vertical plane as the aeroplane, at a height of 3 km above the runway. The light aircraft is travelling with a constant speed of \(2 V \mathrm {~km} / \mathrm { h }\).
At the moment the aeroplane leaves the runway, the light aircraft is at a horizontal distance \(d \mathrm {~km}\) behind the aeroplane. Both aircraft continue to move with the same trajectories due north.

1. Show that the distance, \(D \mathrm {~km}\), between the two aircraft \(t\) hours after the aeroplane leaves the runway satisfies $$D ^ { 2 } = \left( \frac { 6 } { 5 } V t - d \right) ^ { 2 } + \left( \frac { 3 } { 5 } V t - 3 \right) ^ { 2 }$$ Given that the distance between the two aircraft is never less than 2 km ,
2. find the range of possible values for \(d\).

5 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\mathbf { r } = \left( \begin{array} { l } 4

3 Determine whether the lines whose equations are $$\mathbf { r } = ( 1 + 2 \lambda ) \mathbf { i } - \lambda \mathbf { j } + ( 3 + 5 \lambda ) \mathbf { k } \text { and } \mathbf { r } = ( \mu - 1 ) \mathbf { i } + ( 5 - \mu ) \mathbf { j } + ( 2 - 5 \mu ) \mathbf { k }$$ are parallel, intersect or are skew.

1. The line \(l _ { 1 }\) has vector equation \(\mathbf { r } = \left( \begin{array} { r } 5 \\ - 2 \\ 4 \end{array} \right) + \lambda \left( \begin{array} { r } 6 \\ 3 \\ - 1 \end{array} \right)\), where \(\lambda\) is a scalar parameter. The line \(l _ { 2 }\) has vector equation \(\mathbf { r } = \left( \begin{array} { r } 10 \\ 5 \\ - 3 \end{array} \right) + \mu \left( \begin{array} { l } 3 \\ 1 \\ 2 \end{array} \right)\), where \(\mu\) is a scalar parameter.

Justify, giving reasons in each case, whether the lines \(l _ { 1 }\) and \(l _ { 2 }\) are parallel, intersecting or skew.
(6)

3 Show that points \(\mathrm { A } ( 1,4,9 ) , \mathrm { B } ( 0,11,17 )\) and \(\mathrm { C } ( 3 , - 10 , - 7 )\) are collinear.

3 Show that points \(\mathrm { A } ( 1,4,9 ) , \mathrm { B } ( 0,11,17 )\) and \(\mathrm { C } ( 3 , - 10 , - 7 )\) are collinear.

7 Relative to an origin \(O\), the position vectors of points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 0 \\ 2 \\ - 3 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 2 \\ 5 \\ - 2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 3 \\ p \\ q \end{array} \right)$$

1. In the case where \(A B C\) is a straight line, find the values of \(p\) and \(q\).
2. In the case where angle \(B A C\) is \(90 ^ { \circ }\), express \(q\) in terms of \(p\).
3. In the case where \(p = 3\) and the lengths of \(A B\) and \(A C\) are equal, find the possible values of \(q\).

3 A particle is in equilibrium when acted on by the forces \(\left( \begin{array} { r } x \\ - 7 \\ z \end{array} \right) , \left( \begin{array} { r } 4 \\ y \\ - 5 \end{array} \right)\) and \(\left( \begin{array} { r } 5 \\ 4 \\ - 7 \end{array} \right)\), where the units are newtons.

1. Find the values of \(x , y\) and \(z\).
2. Calculate the magnitude of \(\left( \begin{array} { r } 5 \\ 4 \\ - 7 \end{array} \right)\).

2
1 \end{array} \right) , \quad \mathbf { b } = \left( \begin{array} { l } 2
9
0 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { l }

2. Relative to a fixed origin \(O\),
the point \(A\) has position vector \(( 3 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } )\) the point \(B\) has position vector ( \(\mathbf { i } + 2 \mathbf { j } - 4 \mathbf { k }\) )
and the point \(C\) has position vector \(( - \mathbf { i } + \mathbf { j } + a \mathbf { k } )\), where \(a\) is a constant and \(a > 0\).
Given that \(| \overrightarrow { B C } | = \sqrt { 41 }\) a. show that \(a = 2\). \(D\) is the point such that \(A B C D\) forms a parallelogram.
b. Find the position vector of \(D\).

6. Relative to a fixed origin \(O\), the point \(A\) has position vector \(21 \mathbf { i } - 17 \mathbf { j } + 6 \mathbf { k }\) and the point \(B\) has position vector \(25 \mathbf { i } - 14 \mathbf { j } + 18 \mathbf { k }\). The line \(l\) has vector equation $$\mathbf { r } = \left( \begin{array} { r } a
b

2 Find where the line \(\mathbf { r } = \left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right) + \lambda \left( \begin{array} { l } 1 \\ 3 \\ 2 \end{array} \right)\) meets the plane \(2 x + 3 y - 4 z - 5 = 0\).

2 The points \(A\) and \(B\) have position vectors \(3 \mathbf { i } + 2 \mathbf { j }\) and \(4 \mathbf { i } + 2 \mathbf { j } - 5 \mathbf { k }\) respectively.

1. Find the length of \(A B\). Point \(P\) has position vector \(p \mathbf { i } - 3 \mathbf { k }\), where \(p\) is a constant. \(P\) lies on the circumference of a circle of which \(A B\) is a diameter.
2. Find the two possible values of \(p\).