1.10b Vectors in 3D: i,j,k notation

5 A particle moves such that at time \(t\) seconds its acceleration is given by $$( 2 \cos t \mathbf { i } - 5 \sin t \mathbf { j } ) \mathrm { m } \mathrm {~s} ^ { - 2 }$$

1. The mass of the particle is 6 kg . Find the magnitude of the resultant force on the particle when \(t = 0\).
2. When \(t = 0\), the velocity of the particle is \(( 2 \mathbf { i } + 10 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find an expression for the velocity of the particle at time \(t\).

4 A particle moves in a horizontal plane under the action of a single force, \(\mathbf { F }\) newtons. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. At time \(t\) seconds, the position vector, \(\mathbf { r }\) metres, of the particle is given by $$\mathbf { r } = \left( t ^ { 3 } - 3 t ^ { 2 } + 4 \right) \mathbf { i } + \left( 4 t + t ^ { 2 } \right) \mathbf { j }$$

1. Find an expression for the velocity of the particle at time \(t\).
2. The mass of the particle is 3 kg .
   1. Find an expression for \(\mathbf { F }\) at time \(t\).
   2. Find the magnitude of \(\mathbf { F }\) when \(t = 3\).
3. Find the value of \(t\) when \(\mathbf { F }\) acts due north.

1 The velocity of a particle at time \(t\) seconds is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), where $$\mathbf { v } = \left( 4 + 3 t ^ { 2 } \right) \mathbf { i } + ( 12 - 8 t ) \mathbf { j }$$

1. When \(t = 0\), the particle is at the point with position vector \(( 5 \mathbf { i } - 7 \mathbf { j } ) \mathrm { m }\). Find the position vector, \(\mathbf { r }\) metres, of the particle at time \(t\).
2. Find the acceleration of the particle at time \(t\).
3. The particle has mass 2 kg . Find the magnitude of the force acting on the particle when \(t = 1\).

2 A particle, of mass 50 kg , moves on a smooth horizontal plane. A single horizontal force $$\left[ \left( 300 t - 60 t ^ { 2 } \right) \mathbf { i } + 100 \mathrm { e } ^ { - 2 t } \mathbf { j } \right] \text { newtons }$$ acts on the particle at time \(t\) seconds.
The vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors.

1. Find the acceleration of the particle at time \(t\).
2. When \(t = 0\), the velocity of the particle is \(( 7 \mathbf { i } - 4 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). Find the velocity of the particle at time \(t\).
3. Calculate the speed of the particle when \(t = 1\).

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.

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 Two smooth spheres, \(A\) and \(B\), have equal radii and masses \(2 m \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively. The spheres are moving on a smooth horizontal plane. The sphere \(A\) has velocity \(( 3 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it collides with the sphere \(B\), which has velocity \(( 2 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Immediately after the collision, the velocity of the sphere \(B\) is \(( 2 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Find the velocity of \(A\) immediately after the collision.
2. Show that the impulse exerted on \(B\) in the collision is \(( 6 m \mathbf { j } )\) Ns.
3. Find the coefficient of restitution between the two spheres.
4. After the collision, each sphere moves in a straight line with constant speed. Given that the radius of each sphere is 0.05 m , find the time taken, from the collision, until the centres of the spheres are 1.10 m apart.

6 Two smooth spheres, \(A\) and \(B\), have equal radii and masses 4 kg and 2 kg respectively. The sphere \(A\) is moving with velocity \(( 4 \mathbf { i } - 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and the sphere \(B\) is moving with velocity \(( - 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) on the same smooth horizontal surface. The spheres collide when their line of centres is parallel to unit vector \(\mathbf { i }\). The direction of motion of \(B\) is changed through \(90 ^ { \circ }\) by the collision, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{3a1726d9-1b0c-41de-8b43-56019e18aac1-14_332_1184_566_543}

1. Show that the velocity of \(B\) immediately after the collision is \(\left( \frac { 9 } { 2 } \mathbf { i } - 3 \mathbf { j } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\).
2. Find the coefficient of restitution between the spheres.
3. Find the impulse exerted on \(B\) during the collision. State the units of your answer.

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.

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

A coastguard patrol boat \(C\) is moving with constant velocity \(( 8 \mathbf { i } + u \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). Another ship \(S\) is moving with constant velocity \(( 12 \mathbf { i } + 16 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\).

1. Find, in terms of \(u\), the velocity of \(C\) relative to \(S\). At noon, \(S\) is 10 km due west of \(C\).
   If \(C\) is to intercept \(S\),
2. 1. find the value of \(u\).
   2. Using this value of \(u\), find the time at which \(C\) would intercept \(S\). If instead, at noon, \(C\) is moving with velocity \(( 8 \mathbf { i } + 8 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) and continues at this constant velocity,
3. find the distance of closest approach of \(C\) to \(S\).
   

1. A hiker walking due east at a steady speed of \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) notices that the wind appears to come from a direction with bearing 050. At the same time, another hiker moving on a bearing of 320, and also walking at \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\), notices that the wind appears to come from due north.

Find

1. the direction from which the wind is blowing,
2. the wind speed.
   

1. A \(\operatorname { ship } A\) is moving at a constant speed of \(8 \mathrm {~km} \mathrm {~h} \mathrm {~h} ^ { - 1 }\) on a bearing of \(150 ^ { \circ }\). At noon a second ship \(B\) is 6 km from \(A\), on a bearing of \(210 ^ { \circ }\). Ship \(B\) is moving due east at a constant speed. At a later time, \(B\) is \(2 \sqrt { 3 } \mathrm {~km}\) due south of \(A\).

Find

1. the time at which \(B\) will be due east of \(A\),
2. the distance between the ships at that time.

4. A rescue boat, whose maximum speed is \(20 \mathrm {~km} \mathrm {~h} ^ { - 1 }\), receives a signal which indicates that a yacht is in distress near a fixed point \(P\). The rescue boat is 15 km south-west of \(P\). There is a constant current of \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) flowing uniformly from west to east. The rescue boat sets the course needed to get to \(P\) as quickly as possible. Find

1. the course the rescue boat sets,
2. the time, to the nearest minute, to get to \(P\). When the rescue boat arrives at \(P\), the yacht is just visible 4 km due north of \(P\) and is drifting with the current. Find
3. the course that the rescue boat should set to get to the yacht as quickly as possible,
4. the time taken by the rescue boat to reach the yacht from \(P\).

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

Boat \(A\) is moving with velocity ( \(3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) and boat \(B\) is moving with velocity \(( 6 \mathbf { i } - 5 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). Find

1. the magnitude of the velocity of \(A\) relative to \(B\),
2. the direction of the velocity of \(A\) relative to \(B\), giving your answer as a bearing.
   

1. At 10 a.m. two walkers \(A\) and \(B\) are 4 km apart with \(A\) due north of \(B\). Walker \(A\) is moving due east at a constant speed of \(6 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Walker \(B\) is moving with constant speed \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) and walks in the straight line which allows him to pass as close as possible to \(A\).

Find

1. the direction of motion of \(B\), giving your answer as a bearing,
2. the least distance between \(A\) and \(B\),
3. the time when the distance between \(A\) and \(B\) is least.

5. A cyclist riding due north at a steady speed of \(12 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) notices that the wind appears to come from the north-west. At the same time, another cyclist, moving on a bearing of \(120 ^ { \circ }\) and also riding at a steady speed of \(12 \mathrm {~km} \mathrm {~h} ^ { - 1 }\), notices that the wind appears to come from due south. The velocity of the wind is assumed to be constant. Find

1. the wind speed,
2. the direction from which the wind is blowing, giving your answer as a bearing.