1.10e Position vectors: and displacement

2 The unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are directed due east, due north and vertically upwards respectively. Two helicopters, \(A\) and \(B\), are flying with constant velocities of \(( 20 \mathbf { i } - 10 \mathbf { j } + 20 \mathbf { k } ) \mathrm { ms } ^ { - 1 }\) and \(( 30 \mathbf { i } + 10 \mathbf { j } + 10 \mathbf { k } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) respectively. At noon, the position vectors of \(A\) and \(B\) relative to a fixed origin, \(O\), are \(( 8000 \mathbf { i } + 1500 \mathbf { j } + 3000 \mathbf { k } ) \mathrm { m }\) and \(( 2000 \mathbf { i } + 500 \mathbf { j } + 1000 \mathbf { k } ) \mathrm { m }\) respectively.

1. Write down the velocity of \(A\) relative to \(B\).
2. Find the position vector of \(A\) relative to \(B\) at time \(t\) seconds after noon.
3. Find the value of \(t\) when \(A\) and \(B\) are closest together.

2 The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed due east and due north respectively.
Two runners, Albina and Brian, are running on level parkland with constant velocities of \(( 5 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and \(( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) respectively. Initially, the position vectors of Albina and Brian are \(( - 60 \mathbf { i } + 160 \mathbf { j } ) \mathrm { m }\) and \(( 40 \mathbf { i } - 90 \mathbf { j } ) \mathrm { m }\) respectively, relative to a fixed origin in the parkland.

1. Write down the velocity of Brian relative to Albina.
2. Find the position vector of Brian relative to Albina \(t\) seconds after they leave their initial positions.
3. Hence determine whether Albina and Brian will collide if they continue running with the same velocities.

4 The unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are directed east, north and vertically upwards respectively. At time \(t = 0\), the position vectors of two small aeroplanes, \(A\) and \(B\), relative to a fixed origin \(O\) are \(( - 60 \mathbf { i } + 30 \mathbf { k } ) \mathrm { km }\) and \(( - 40 \mathbf { i } + 10 \mathbf { j } - 10 \mathbf { k } ) \mathrm { km }\) respectively. The aeroplane \(A\) is flying with constant velocity \(( 250 \mathbf { i } + 50 \mathbf { j } - 100 \mathbf { k } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) and the aeroplane \(B\) is flying with constant velocity \(( 200 \mathbf { i } + 25 \mathbf { j } + 50 \mathbf { k } ) \mathrm { km } \mathrm { h } ^ { - 1 }\).

1. Write down the position vectors of \(A\) and \(B\) at time \(t\) hours.
2. Show that the position vector of \(A\) relative to \(B\) at time \(t\) hours is \(( ( - 20 + 50 t ) \mathbf { i } + ( - 10 + 25 t ) \mathbf { j } + ( 40 - 150 t ) \mathbf { k } ) \mathrm { km }\).
3. Show that \(A\) and \(B\) do not collide.
4. Find the value of \(t\) when \(A\) and \(B\) are closest together.
   \includegraphics[max width=\textwidth, alt={}]{01071eb0-2c48-4028-8cd3-6021ce86d7e5-13_2484_1709_223_153}

4 The unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are directed due east, due north and vertically upwards respectively. A helicopter, \(A\), is travelling in the direction of the vector \(- 2 \mathbf { i } + 3 \mathbf { j } + 6 \mathbf { k }\) with constant speed \(140 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Another helicopter, \(B\), is travelling in the direction of the vector \(2 \mathbf { i } - \mathbf { j } + 2 \mathbf { k }\) with constant speed \(60 \mathrm {~km} \mathrm {~h} ^ { - 1 }\).

1. Find the velocity of \(A\) relative to \(B\).
2. Initially, the position vectors of \(A\) and \(B\) are \(( 4 \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } ) \mathrm { km }\) and \(( - 3 \mathbf { i } + 6 \mathbf { j } + 3 \mathbf { k } ) \mathrm { km }\) respectively, relative to a fixed origin. Write down the position vector of \(A\) relative to \(B , t\) hours after they leave their initial positions.
3. Find the distance between \(A\) and \(B\) when they are closest together.
   
   \includegraphics[max width=\textwidth, alt={}]{0590950d-145c-4ae2-bc3c-f61a9433d158-10_2486_1714_221_153}
   \includegraphics[max width=\textwidth, alt={}]{0590950d-145c-4ae2-bc3c-f61a9433d158-11_2486_1714_221_153}

6 At noon, two ships, \(A\) and \(B\), are a distance of 12 km apart, with \(B\) on a bearing of \(065 ^ { \circ }\) from \(A\). The ship \(B\) travels due north at a constant speed of \(10 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). The ship \(A\) travels at a constant speed of \(18 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{a90a2de3-5cc0-4e87-b29a-2562f86eee17-16_492_585_445_738}

1. Find the direction in which \(A\) should travel in order to intercept \(B\). Give your answer as a bearing.
2. In fact, the ship \(A\) actually travels on a bearing of \(065 ^ { \circ }\).
   1. Find the distance between the ships when they are closest together.
   2. Find the time when the ships are closest together.

7 From an aircraft \(A\), a helicopter \(H\) is observed 20 km away on a bearing of \(120 ^ { \circ }\). The helicopter \(H\) is travelling horizontally with a constant speed \(240 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on a bearing of \(340 ^ { \circ }\). The aircraft \(A\) is travelling with constant speed \(v _ { A } \mathrm {~km} \mathrm {~h} ^ { - 1 }\) in a straight line and at the same altitude as \(H\). \includegraphics[max width=\textwidth, alt={}, center]{3a1726d9-1b0c-41de-8b43-56019e18aac1-18_774_801_504_607}

1. Given that \(v _ { A } = 200\) :
   1. find a bearing, to one decimal place, on which \(A\) could travel in order to intercept \(H\);
   2. find the time, in minutes, that it would take \(A\) to intercept \(H\) on this bearing.
2. Given that \(v _ { A } = 150\), find the bearing on which \(A\) should travel in order to approach \(H\) as closely as possible. Give your answer to one decimal place.
   \includegraphics[max width=\textwidth, alt={}]{3a1726d9-1b0c-41de-8b43-56019e18aac1-20_2253_1691_221_153}

4 Two boats, \(A\) and \(B\), are moving on straight courses with constant speeds. At noon, \(A\) and \(B\) have position vectors \(( \mathbf { i } + 2 \mathbf { j } ) \mathrm { km }\) and \(( - \mathbf { i } + \mathbf { j } ) \mathrm { km }\) respectively relative to a lighthouse. Thirty minutes later, the position vectors of \(A\) and \(B\) are ( \(- \mathbf { i } + 3 \mathbf { j }\) ) km and \(( 2 \mathbf { i } - \mathbf { j } ) \mathrm { km }\) respectively relative to the lighthouse.

1. Find the velocity of \(A\) relative to \(B\) in the form \(( m \mathbf { i } + n \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\), where \(m\) and \(n\) are integers.
2. The position vector of \(A\) relative to \(B\) at time \(t\) hours after noon is \(\mathbf { r } \mathrm { km }\). Show that $$\mathbf { r } = ( 2 - 10 t ) \mathbf { i } + ( 1 + 6 t ) \mathbf { j }$$
3. Determine the value of \(t\) when \(A\) and \(B\) are closest together.
4. Find the shortest distance between \(A\) and \(B\).

2. Ship \(A\) is steaming on a bearing of \(060 ^ { \circ }\) at \(30 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) and at 9 a.m. it is 20 km due west of a second ship \(B\). Ship \(B\) steams in a straight line.

1. Find the least speed of \(B\) if it is to intercept \(A\). Given that the speed of \(B\) is \(24 \mathrm {~km} \mathrm {~h} ^ { - 1 }\),
2. find the earliest time at which it can intercept \(A\).

5. At time \(t = 0\) particles \(P\) and \(Q\) start simultaneously from points which have position vectors \(( \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } ) \mathrm { m }\) and \(( - \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) \mathrm { m }\) respectively, relative to a fixed origin \(O\). The velocities of \(P\) and \(Q\) are \(( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and \(( 2 \mathbf { i } + \mathbf { k } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) respectively.

1. Show that \(P\) and \(Q\) collide and find the position vector of the point at which they collide. A third particle \(R\) moves in such a way that its velocity relative to \(P\) is parallel to the vector ( \(- 5 \mathbf { i } + 4 \mathbf { j } - \mathbf { k }\) ) and its velocity relative to \(Q\) is parallel to the vector \(( - 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } )\). Given that all three particles collide simultaneously, find
2. 1. the velocity of \(R\),
   2. the position vector of \(R\) at time \(t = 0\).

2. \begin{figure}[h] \end{figure} A man, who rows at a speed \(v\) through still water, rows across a river which flows at a speed \(u\). The man sets off from the point \(A\) on one bank and wishes to land at the point \(B\) on the opposite bank, where \(A B\) is perpendicular to both banks, as shown in Fig. 1.

1. Show that, for this to be possible, \(v > u\). Given that \(v < u\) and that he rows from \(A\) so as to reach a point \(C\), on the opposite bank, which is as close to \(B\) as possible,
2. find, in terms of \(u\) and \(v\), the ratio of \(B C\) to the width of the river.
   (5)

3. A man walks due north at a constant speed \(u\) and the wind seems to him to be blowing from the direction \(30 ^ { \circ }\) east of north. On his return journey, when he is walking at the same speed \(u\) due south, the wind seems to him to be blowing from the direction \(30 ^ { \circ }\) south of east. Assuming that the velocity, \(\mathbf { w }\), of the wind relative to the earth is constant, find

1. the magnitude of \(\mathbf { w }\), in terms of \(u\),
2. the direction of \(\mathbf { w }\).

1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively.]

An aeroplane makes a journey from a point \(P\) to a point \(Q\) which is due east of \(P\). The wind velocity is \(w ( \cos \theta \mathbf { i } + \sin \theta \mathbf { j } )\), where \(w\) is a positive constant. The velocity of the aeroplane relative to the wind is \(v ( \cos \phi \mathbf { i } - \sin \phi \mathbf { j } )\), where \(v\) is a constant and \(v > w\). Given that \(\theta\) and \(\phi\) are both acute angles,

1. show that \(v \sin \phi = w \sin \theta\),
2. find, in terms of \(v , w\) and \(\theta\), the speed of the aeroplane relative to the ground.

3. At noon, two boats \(A\) and \(B\) are 6 km apart with \(A\) due east of \(B\). Boat \(B\) is moving due north at a constant speed of \(13 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Boat \(A\) is moving with constant speed \(12 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) and sets a course so as to pass as close as possible to boat \(B\). Find

1. the direction of motion of \(A\), giving your answer as a bearing,
2. the time when the boats are closest,
3. the shortest distance between the boats.

5 Two aircraft \(A\) and \(B\) are flying horizontally at the same height. \(A\) has constant velocity \(240 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction with bearing \(025 ^ { \circ }\), and \(B\) has constant velocity \(185 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction with bearing \(310 ^ { \circ }\).

1. Find the magnitude and direction of the velocity of \(A\) relative to \(B\). Initially \(A\) is 4500 m due west of \(B\). For the instant during the subsequent motion when \(A\) and \(B\) are closest together, find
2. the distance between \(A\) and \(B\),
3. the bearing of \(A\) from \(B\).

4 A boat \(A\) has constant velocity \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction with bearing \(110 ^ { \circ }\). A boat \(B\), which is initially 250 m due south of \(A\), moves with constant speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction which takes it as close as possible to \(A\).

1. Find the bearing of the direction in which \(B\) moves.
2. Find the shortest distance between \(A\) and \(B\) in the subsequent motion.

5 A ship \(S\) is travelling with constant speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a course with bearing \(345 ^ { \circ }\). A patrol boat \(B\) spots the ship \(S\) when \(S\) is 2400 m from \(B\) on a bearing of \(050 ^ { \circ }\). The boat \(B\) sets off in pursuit, travelling with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line.

1. Given that \(v = 16\), find the bearing of the course which \(B\) should take in order to intercept \(S\), and the time taken to make the interception.
2. Given instead that \(v = 10\), find the bearing of the course which \(B\) should take in order to get as close as possible to \(S\). \includegraphics[max width=\textwidth, alt={}, center]{181fad74-6e60-4435-a176-3edff5062c32-4_337_954_278_544} A uniform rod \(A B\) has mass \(m\) and length \(2 a\). The point \(P\) on the rod is such that \(A P = \frac { 2 } { 3 } a\). The rod is placed in a horizontal position perpendicular to the edge of a rough horizontal table, with \(A P\) in contact with the table and \(P B\) overhanging the edge. The rod is released from rest in this position. When it has rotated through an angle \(\theta\), and no slipping has occurred at \(P\), the normal reaction acting on the rod at \(P\) is \(R\) and the frictional force is \(F\) (see diagram).
3. Show that the angular acceleration of the rod is \(\frac { 3 g \cos \theta } { 4 a }\).
4. Find the angular speed of the rod, in terms of \(a , g\) and \(\theta\).
5. Find \(F\) and \(R\) in terms of \(m , g\) and \(\theta\).
6. Given that the coefficient of friction between the rod and the edge of the table is \(\mu\), show that the rod is on the point of slipping at \(P\) when \(\tan \theta = \frac { 1 } { 2 } \mu\). \includegraphics[max width=\textwidth, alt={}, center]{181fad74-6e60-4435-a176-3edff5062c32-5_677_624_269_753} A smooth circular wire, with centre \(O\) and radius \(a\), is fixed in a vertical plane. The highest point on the wire is \(A\) and the lowest point on the wire is \(B\). A small ring \(R\) of mass \(m\) moves freely along the wire. A light elastic string, with natural length \(a\) and modulus of elasticity \(\frac { 1 } { 2 } m g\), has one end attached to \(A\) and the other end attached to \(R\). The string \(A R\) makes an angle \(\theta\) (measured anticlockwise) with the downward vertical, so that \(O R\) makes an angle \(2 \theta\) with the downward vertical (see diagram). You may assume that the string does not become slack.
7. Taking \(A\) as the level for zero gravitational potential energy, show that the total potential energy \(V\) of the system is given by $$V = m g a \left( \frac { 1 } { 4 } - \cos \theta - \cos ^ { 2 } \theta \right) .$$
8. Show that \(\theta = 0\) is the only position of equilibrium.
9. By differentiating the energy equation with respect to time \(t\), show that $$\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - \frac { g } { 4 a } \sin \theta ( 1 + 2 \cos \theta ) .$$
10. Deduce the approximate period of small oscillations about the equilibrium position \(\theta = 0\).

1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors.]

A small smooth ring of mass 0.5 kg moves along a smooth horizontal wire. The only forces acting on the ring are its weight, the normal reaction from the wire, and a constant force \(( 5 \mathbf { i } + \mathbf { j } - 3 \mathbf { k } ) \mathrm { N }\). The ring is initially at rest at the point with position vector \(( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\), relative to a fixed origin. Find the speed of the ring as it passes through the point with position vector \(( 3 \mathbf { i } + \mathbf { k } ) \mathrm { m }\).

1. Three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) are acting on an object of mass 3 kg such that

$$\begin{aligned} & \mathbf { F } _ { 1 } = ( \mathbf { i } + 8 c \mathbf { j } + 11 c \mathbf { k } ) \mathrm { N } , \\ & \mathbf { F } _ { 2 } = ( - 14 \mathbf { i } - c \mathbf { j } - 12 \mathbf { k } ) \mathrm { N } , \\ & \mathbf { F } _ { 3 } = ( ( 15 c + 1 ) \mathbf { i } + 2 c \mathbf { j } - 5 c \mathbf { k } ) \mathrm { N } , \end{aligned}$$ where \(c\) is a constant. The acceleration of the object is parallel to the vector \(( \mathbf { i } + \mathbf { j } )\).

1. Find the value of the constant \(c\) and hence show that the acceleration of the object is \(( 6 \mathbf { i } + 6 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\).
2. When \(t = 0\) seconds, the object has position vector \(\mathbf { r } _ { 0 } \mathrm {~m}\) and is moving with velocity \(( - 17 \mathbf { i } + 8 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). When \(t = 4\) seconds, the object has position vector \(( - 13 \mathbf { i } + 84 \mathbf { j } ) \mathrm { m }\). Find the vector \(\mathbf { r } _ { 0 }\).

5 Points \(A , B\) and \(C\) have position vectors \(\left( \begin{array} { l } 1 \\ 2 \\ 3 \end{array} \right) , \left( \begin{array} { c } 2 \\ - 1 \\ 5 \end{array} \right)\) and \(\left( \begin{array} { c } - 4 \\ 0 \\ 3 \end{array} \right)\) respectively.

1. Find the exact distance between the midpoint of \(A B\) and the midpoint of \(B C\). Point \(D\) has position vector \(\left( \begin{array} { c } x \\ - 6 \\ z \end{array} \right)\) and the line \(C D\) is parallel to the line \(A B\).
2. Find all the possible pairs of \(x\) and \(z\).

8 The points \(A , B\) and \(C\) have position vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\), relative to an origin \(O\), in three dimensions. The figure \(O A P B S C T U\) is a cuboid, with vertices labelled as in the following diagram. \(M\) is the midpoint of \(A U\). \includegraphics[max width=\textwidth, alt={}, center]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-5_557_1221_2087_420}

7. Two ships \(P\) and \(Q\) are travelling at night with constant velocities. At midnight, \(P\) is at the point with position vector \(( 20 \mathbf { i } + 10 \mathbf { j } ) \mathrm { km }\) relative to a fixed origin \(O\). At the same time, \(Q\) is at the point with position vector \(( 14 \mathbf { i } - 6 \mathbf { j } ) \mathrm { km }\). Three hours later, \(P\) is at the point with position vector \(( 29 \mathbf { i } + 34 \mathbf { j } ) \mathrm { km }\). The ship \(Q\) travels with velocity \(12 \mathbf { j } \mathrm {~km} \mathrm {~h} ^ { - 1 }\). At time \(t\) hours after midnight, the position vectors of \(P\) and \(Q\) are \(\mathbf { p } \mathrm { km }\) and \(\mathbf { q } \mathrm { km }\) respectively. Find

1. the velocity of \(P\), in terms of \(\mathbf { i }\) and \(\mathbf { j }\),
2. expressions for \(\mathbf { p }\) and \(\mathbf { q }\), in terms of \(t\), i and \(\mathbf { j }\). At time \(t\) hours after midnight, the distance between \(P\) and \(Q\) is \(d \mathrm {~km}\).
3. By finding an expression for \(\overrightarrow { P Q }\), show that $$d ^ { 2 } = 25 t ^ { 2 } - 92 t + 292$$ Weather conditions are such that an observer on \(P\) can only see the lights on \(Q\) when the distance between \(P\) and \(Q\) is 15 km or less. Given that when \(t = 1\), the lights on \(Q\) move into sight of the observer,
4. find the time, to the nearest minute, at which the lights on \(Q\) move out of sight of the observer.
   1. In taking off, an aircraft moves on a straight runway \(A B\) of length 1.2 km . The aircraft moves from \(A\) with initial speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It moves with constant acceleration and 20 s later it leaves the runway at \(C\) with speed \(74 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
   2. the acceleration of the aircraft,
   3. the distance \(B C\).
   4. Two small steel balls \(A\) and \(B\) have mass 0.6 kg and 0.2 kg respectively. They are moving towards each other in opposite directions on a smooth horizontal table when they collide directly. Immediately before the collision, the speed of \(A\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision, the direction of motion of \(A\) is unchanged and the speed of \(B\) is twice the speed of \(A\). Find
   5. the speed of \(A\) immediately after the collision,
   6. the magnitude of the impulse exerted on \(B\) in the collision.
      
   \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-018_282_707_278_699}
   \end{figure}

9 The points \(A\) and \(B\) lie on the line \(l _ { 1 }\) and have coordinates \(( 2,5,1 )\) and \(( 4,1 , - 2 )\) respectively.

1. 1. Find the vector \(\overrightarrow { A B }\).
   2. Find a vector equation of the line \(l _ { 1 }\), with parameter \(\lambda\).
2. The line \(l _ { 2 }\) has equation \(\mathbf { r } = \left[ \begin{array} { r } 1 \\ - 3 \\ - 1 \end{array} \right] + \mu \left[ \begin{array} { r } 1 \\ 0 \\ - 2 \end{array} \right]\).
   1. Show that the point \(P ( - 2 , - 3,5 )\) lies on \(l _ { 2 }\).
   2. The point \(Q\) lies on \(l _ { 1 }\) and is such that \(P Q\) is perpendicular to \(l _ { 2 }\). Find the coordinates of \(Q\).

8 The points \(A\) and \(B\) have coordinates \(( 2,1 , - 1 )\) and \(( 3,1 , - 2 )\) respectively. The angle \(O B A\) is \(\theta\), where \(O\) is the origin.

1. 1. Find the vector \(\overrightarrow { A B }\).
   2. Show that \(\cos \theta = \frac { 5 } { 2 \sqrt { 7 } }\).
2. The point \(C\) is such that \(\overrightarrow { O C } = 2 \overrightarrow { O B }\). The line \(l\) is parallel to \(\overrightarrow { A B }\) and passes through the point \(C\). Find a vector equation of \(l\).
3. The point \(D\) lies on \(l\) such that angle \(O D C = 90 ^ { \circ }\). Find the coordinates of \(D\).

8 The points \(A , B\) and \(C\) have coordinates \(( 2 , - 1 , - 5 ) , ( 0,5 , - 9 )\) and \(( 9,2,3 )\) respectively. The line \(l\) has equation \(\mathbf { r } = \left[ \begin{array} { r } 2 \\ - 1 \\ - 5 \end{array} \right] + \lambda \left[ \begin{array} { r } 1 \\ - 3 \\ 2 \end{array} \right]\).

1. Verify that the point \(B\) lies on the line \(l\).
2. Find the vector \(\overrightarrow { B C }\).
3. The point \(D\) is such that \(\overrightarrow { A D } = 2 \overrightarrow { B C }\).
   1. Show that \(D\) has coordinates \(( 20 , - 7,19 )\).
   2. The point \(P\) lies on \(l\) where \(\lambda = p\). The line \(P D\) is perpendicular to \(l\). Find the value of \(p\).

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

1. Show that the distance between the points \(A\) and \(B\) is \(3 \sqrt { 3 }\).
2. The line \(A B\) makes an acute angle \(\theta\) with \(l\). Show that \(\cos \theta = \frac { 7 } { 9 }\).
3. The point \(P\) on the line \(l\) is where \(\lambda = p\).
   1. Show that $$\overrightarrow { A P } \cdot \left[ \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right] = 7 + 3 p$$
   2. Hence find the coordinates of the foot of the perpendicular from the point \(A\) to the line \(l\).