1.10f Distance between points: using position vectors

\(\mathbf { 7 } \quad\) 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 \(B\).

6 The points \(A , B\) and \(C\) have coordinates \(( 3 , - 2,4 ) , ( 1 , - 5,6 )\) and \(( - 4,5 , - 1 )\) respectively. The line \(l\) passes through \(A\) and has equation \(\mathbf { r } = \left[ \begin{array} { r } 3 \\ - 2 \\ 4 \end{array} \right] + \lambda \left[ \begin{array} { r } 7 \\ - 7 \\ 5 \end{array} \right]\).

1. Show that the point \(C\) lies on the line \(l\).
2. Find a vector equation of the line that passes through points \(A\) and \(B\).
3. The point \(D\) lies on the line through \(A\) and \(B\) such that the angle \(C D A\) is a right angle. Find the coordinates of \(D\).
4. The point \(E\) lies on the line through \(A\) and \(B\) such that the area of triangle \(A C E\) is three times the area of triangle \(A C D\). Find the coordinates of the two possible positions of \(E\).

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 \(E\).
   [0pt] [4 marks]

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

5. Relative to a fixed origin, the points \(A , B\) and \(C\) have position vectors ( \(2 \mathbf { i } - \mathbf { j } + 6 \mathbf { k }\) ), \(( 5 \mathbf { i } - 4 \mathbf { j } )\) and \(( 7 \mathbf { i } - 6 \mathbf { j } - 4 \mathbf { k } )\) respectively.

1. Show that \(A , B\) and \(C\) all lie on a single straight line.
2. Write down the ratio \(A B : B C\) The point \(D\) has position vector \(( 3 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } )\).
3. Show that \(A D\) is perpendicular to \(B D\).
4. Find the exact area of triangle \(A B D\).
   5. continued

5. Two dogs, Fido and Growler, are playing in a field. Fido is moving in a straight line so that at time \(t\) his position vector relative to a fixed origin, \(O\), is given by \([ ( 2 t - 3 ) \mathbf { i } + t \mathbf { j } ]\) metres. Growler is stationary at the point with position vector \(( 2 \mathbf { i } + 5 \mathbf { j } )\) metres, where \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal perpendicular unit vectors.

1. Find the displacement vector of Fido from Growler in terms of \(t\).
2. Find the value of \(t\) for which the two dogs are closest.
3. Find the minimum distance between the two dogs.

2 In this question, positions are given relative to a fixed origin, O. The \(x\)-direction is east and the \(y\)-direction north; distances are measured in kilometres. Two boats, the Rosemary and the Sage, are having a race between two points A and B.
The position vector of the Rosemary at time \(t\) hours after the start is given by $$\mathbf { r } = \binom { 3 } { 2 } + \binom { 6 } { 8 } t , \text { where } 0 \leqslant t \leqslant 2 .$$ The Rosemary is at point A when \(t = 0\), and at point B when \(t = 2\).

1. Find the distance AB .
2. Show that the Rosemary travels at constant velocity. The position vector of the Sage is given by $$\mathbf { r } = \binom { 3 ( 2 t + 1 ) } { 2 \left( 2 t ^ { 2 } + 1 \right) }$$
3. Plot the points A and B . Draw the paths of the two boats for \(0 \leqslant t \leqslant 2\).
4. What can you say about the result of the race?
5. Find the speed of the Sage when \(t = 2\). Find also the direction in which it is travelling, giving your answer as a compass bearing, to the nearest degree.
6. Find the displacement of the Rosemary from the Sage at time \(t\) and hence calculate the greatest distance between the boats during the race.

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 \(B\).

1. Particles \(P\) and \(Q\) move in a plane with constant velocities. At time \(t = 0\) the position vectors of \(P\) and \(Q\), relative to a fixed point \(O\) in the plane, are \(( 16 \mathbf { i } - 12 \mathbf { j } ) \mathrm { m }\) and \(( - 5 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m }\) respectively. The velocity of \(P\) is \(( \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(Q\) is \(( 2 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)

Find the shortest distance between \(P\) and \(Q\) in the subsequent motion.

3. Two straight horizontal roads cross at right angles at the point \(X\). A girl is running, with constant speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), due north towards \(X\) on one road. A car is travelling, with constant speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), due west towards \(X\) on the other road.

1. Find the magnitude and direction of the velocity of the car relative to the girl, giving the direction as a bearing.
   (6) At noon the girl is 150 m south of \(X\) and the car is 800 m east of \(X\).
2. Find the shortest distance between the car and the girl during the subsequent motion.

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

A ship \(A\) has constant velocity \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { kmh } ^ { - 1 }\) and a ship \(B\) has constant velocity \(( - \mathbf { i } + 3 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). At noon, the position vectors of the ships \(A\) and \(B\) with respect to a fixed origin \(O\) are \(( - 2 \mathbf { i } + \mathbf { j } ) \mathrm { km }\) and \(( 5 \mathbf { i } - 2 \mathbf { j } ) \mathrm { km }\) respectively. Find

1. the time at which the two ships are closest together,
2. the length of time for which ship \(A\) is within 2 km of ship \(B\).

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).

1. Show that the angular acceleration of the rod is \(\frac { 3 g \cos \theta } { 4 a }\).
2. Find the angular speed of the rod, in terms of \(a , g\) and \(\theta\).
3. Find \(F\) and \(R\) in terms of \(m , g\) and \(\theta\).
4. 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.

1. Taking \(A\) as the level for zero gravitational potential energy, show that the total potential energy \(V\) of the system is given by $$V = m g a \left( \frac { 1 } { 4 } - \cos \theta - \cos ^ { 2 } \theta \right) .$$
2. Show that \(\theta = 0\) is the only position of equilibrium.
3. By differentiating the energy equation with respect to time \(t\), show that $$\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - \frac { g } { 4 a } \sin \theta ( 1 + 2 \cos \theta ) .$$
4. 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 perpendicular unit vectors in a horizontal plane.]

A bead \(P\) of mass 0.4 kg is threaded on a smooth straight horizontal wire. The wire lies along the line with vector equation \(\mathbf { r } = ( \mathbf { i } + 2 \mathbf { j } ) + \lambda ( - 2 \mathbf { i } + 3 \mathbf { j } )\). The bead is initially at rest at the point \(A\) with position vector \(( - \mathbf { i } + 5 \mathbf { j } ) \mathrm { m }\). A constant horizontal force \(( 0.5 \mathbf { i } + \mathbf { j } ) \mathrm { N }\) acts on \(P\) and moves it along the wire to the point \(B\). At \(B\) the speed of \(P\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the position vector of \(B\).

1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal, \(x - y\) plane.]

A bead \(P\) of mass 0.08 kg is threaded on a smooth straight horizontal wire which lies along the line with equation \(y = 2 x - 1\). The unit of length on both axes is the metre. Initially the bead is at rest at the point \(( a , b )\). A force \(( 6 \mathbf { i } - 2 \mathbf { j } ) \mathrm { N }\) acts on \(P\) and moves it along the wire so that \(P\) passes through the point \(( 5,9 )\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the value of \(a\) and the value of \(b\).