1.10a Vectors in 2D: i,j notation and column vectors

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.

3. When a man walks due West at a constant speed of \(4 \mathrm {~km} \mathrm {~h} ^ { - 1 }\), the wind appears to be blowing from due South. When he runs due North at a constant speed of \(8 \mathrm {~km} \mathrm {~h} ^ { - 1 }\), the speed of the wind appears to be \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\).
The velocity of the wind relative to the Earth is constant with magnitude \(w \mathrm {~km} \mathrm {~h} ^ { - 1 }\).
Find the two possible values of \(w\).

4 \includegraphics[max width=\textwidth, alt={}, center]{6e3d5f5e-7ffa-4111-903d-468fb4d20192-2_364_1313_1224_376} An unidentified aircraft \(U\) is flying horizontally with constant velocity \(250 \mathrm {~ms} ^ { - 1 }\) in the direction with bearing \(040 ^ { \circ }\). Two spotter planes \(P\) and \(Q\) are flying horizontally at the same height as \(U\), and at one instant \(P\) is 15000 m due west of \(U\), and \(Q\) is 15000 m due east of \(U\) (see diagram).

1. Plane \(P\) is flying with constant velocity \(210 \mathrm {~ms} ^ { - 1 }\) in the direction with bearing \(070 ^ { \circ }\).

1 Alan is running in a straight line on a bearing of \(090 ^ { \circ }\) at a constant speed of \(4 \mathrm {~ms} ^ { - 1 }\). Ben sees Alan when they are 50 m apart and Alan is on a bearing of \(060 ^ { \circ }\) from Ben. Ben sets off immediately to intercept Alan by running at a constant speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Calculate the bearing on which Ben should run to intercept Alan.
2. Calculate the magnitude of the velocity of Ben relative to Alan and find the time it takes, from the moment Ben sees Alan, for Ben to intercept Alan.

3 Two planes, \(A\) and \(B\), flying at the same altitude, are participating in an air show. Initially the planes are 400 m apart and plane \(B\) is on a bearing of \(130 ^ { \circ }\) from plane \(A\). Plane \(A\) is moving due south with a constant speed of \(75 \mathrm {~ms} ^ { - 1 }\). Plane \(B\) is moving at a constant speed of \(40 \mathrm {~ms} ^ { - 1 }\) and has set a course to get as close as possible to \(A\).

1. Find the bearing of the course set by \(B\) and the shortest distance between the two planes in the subsequent motion.
2. Find the total distance travelled by \(A\) and \(B\) from the instant when they are initially 400 m apart to the point of their closest approach.

2 A ship \(S\) is travelling with constant speed \(5 \mathrm {~ms} ^ { - 1 }\) on a course with bearing \(325 ^ { \circ }\). A second ship \(T\) observes \(S\) when \(S\) is 9500 m from \(T\) on a bearing of \(060 ^ { \circ }\) from \(T\). Ship \(T\) sets off in pursuit, travelling with constant speed \(8.5 \mathrm {~ms} ^ { - 1 }\) in a straight line.

1. Find the bearing of the course which \(T\) should take in order to intercept \(S\).
2. Find the distance travelled by \(S\) from the moment that \(T\) sets off in pursuit until the point of interception.

1. A bead is threaded on a straight wire. The vector equation of the wire is

$$\mathbf { r } = \mathbf { i } - 3 \mathbf { j } + \mathbf { k } + t ( 2 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } )$$ where the unit of length is the metre. The bead is moved from a point \(A\) on the wire through a distance of 6 m along the wire to a point \(B\) by a force \(\mathbf { F } = ( 7 \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k } ) \mathrm { N }\). Find the magnitude of the work done by \(\mathbf { F }\) in moving the bead from \(A\) to \(B\).
(Total 4 marks)

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 ( \(5 \mathbf { i } + \mathbf { j } - 3 \mathbf { k }\) ) 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. In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane and \(\mathbf { k }\) is a unit vector vertically upwards.

A small smooth ring of mass 0.1 kg is threaded onto a smooth horizontal wire which is parallel to \(( \mathbf { i } + 2 \mathbf { j } )\). The only forces acting on the ring are its weight, the normal reaction from the wire and a constant force \(( \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k } )\) N. The ring starts from rest at the point \(A\) on the wire, whose position vector relative to a fixed origin is \(( 2 \mathbf { i } - 2 \mathbf { j } - 3 \mathbf { k } ) \mathrm { m }\), and passes through the point \(B\) with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the position vector of \(B\).
(6)

3. \section*{Figure 1} Figure 1 shows a box in the shape of a cuboid \(P Q R S T U V W\) where \(\overrightarrow { P Q } = 3 \mathbf { i }\) metres, \(\overrightarrow { P S } = 4 \mathbf { j }\) metres and \(\overrightarrow { P T } = 3 \mathbf { k }\) metres. A force \(( 4 \mathbf { i } - 2 \mathbf { j } ) \mathrm { N }\) acts at \(Q\), a force \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }\) acts at \(R\), a force \(( - 2 \mathbf { j } + \mathbf { k } ) \mathrm { N }\) acts at \(T\), and a force \(( 2 \mathbf { j } + \mathbf { k } ) \mathrm { N }\) acts at \(W\). Given that these are the only forces acting on the box, find

1. the resultant force acting on the box,
2. the resultant vector moment about \(P\) of the four forces acting on the box. When an additional force \(\mathbf { F }\) acts on the box at a point \(X\) on the edge \(P S\), the box is in equilibrium.
3. Find \(\mathbf { F }\).
4. Find the length of \(P X\).

2. Two constant forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) are the only forces acting on a particle \(P\) of mass 2 kg . The particle is initially at rest at the point \(A\) with position vector \(( - 2 \mathbf { i } - \mathbf { j } - 4 \mathbf { k } ) \mathrm { m }\). Four seconds later, \(P\) is at the point \(B\) with position vector \(( 6 \mathbf { i } - 5 \mathbf { j } + 8 \mathbf { k } ) \mathrm { m }\). Given that \(\mathbf { F } _ { 1 } = ( 12 \mathbf { i } - 4 \mathbf { j } + 6 \mathbf { k } ) \mathrm { N }\), find

1. \(\mathbf { F } _ { 2 }\),
2. the work done on \(P\) as it moves from \(A\) to \(B\).

6. A particle \(P\) of mass 2 kg moves in the \(x - y\) plane. At time \(t\) seconds its position vector is \(\mathbf { r }\) metres. When \(t = 0\), the position vector of \(P\) is \(\mathbf { i }\) metres and the velocity of \(P\) is ( \(- \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The vector \(\mathbf { r }\) satisfies the differential equation $$\frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} \mathbf { r } } { \mathrm {~d} t } + 2 \mathbf { r } = \mathbf { 0 }$$

1. Find \(\mathbf { r }\) in terms of \(t\).
2. Show that the speed of \(P\) at time \(t\) is \(\mathrm { e } ^ { - t } \sqrt { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find, in terms of e, the loss of kinetic energy of \(P\) in the interval \(t = 0\) to \(t = 1\).

1. Two constant forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) are the only forces acting on a particle. \(\mathbf { F } _ { 1 }\) has magnitude 9 N and acts in the direction of \(2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } . \mathbf { F } _ { 2 }\) has magnitude 18 N and acts in the direction of \(\mathbf { i } + 8 \mathbf { j } - 4 \mathbf { k }\).

Find the total work done by the two forces in moving the particle from the point with position vector \(( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\) to the point with position vector \(( 3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) \mathrm { m }\).
(Total 6 marks)

2. At time \(t\) seconds the position vector of a particle \(P\), relative to a fixed origin \(O\), is \(\mathbf { r }\) metres, where \(\mathbf { r }\) satisfies the differential equation $$\frac { \mathrm { d } \mathbf { r } } { \mathrm {~d} t } + 2 \mathbf { r } = 3 \mathrm { e } ^ { - t } \mathbf { j }$$ Given that \(\mathbf { r } = 2 \mathbf { i } - \mathbf { j }\) when \(t = 0\), find \(\mathbf { r }\) in terms of \(t\).
(Total 7 marks)

3. A system of forces acting on a rigid body consists of two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) acting at a point \(A\) of the body, together with a couple of moment \(\mathbf { G } . \mathbf { F } _ { 1 } = ( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( - 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) N\). The position vector of the point \(A\) is \(( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\) and \(\mathbf { G } = ( 7 \mathbf { i } - 3 \mathbf { j } + 8 \mathbf { k } ) \mathrm { Nm }\). Given that the system is equivalent to a single force \(\mathbf { R }\),

1. find \(\mathbf { R }\),
2. find a vector equation for the line of action of \(\mathbf { R }\).
   (Total 9 marks) \section*{4.} \section*{Figure 1}
   \includegraphics[max width=\textwidth, alt={}]{43ce237f-c8e4-428a-b8cd-04673e62abb9-3_896_515_276_772}
   A thin uniform rod \(P Q\) has mass \(m\) and length \(3 a\). A thin uniform circular disc, of mass \(m\) and radius \(a\), is attached to the rod at \(Q\) in such a way that the rod and the diameter \(Q R\) of the disc are in a straight line with \(P R = 5 a\). The rod together with the disc form a composite body, as shown in Figure 1. The body is free to rotate about a fixed smooth horizontal axis \(L\) through \(P\), perpendicular to \(P Q\) and in the plane of the disc.

1. A bead of mass 0.5 kg is threaded on a smooth straight wire. The only forces acting on the bead are a constant force ( \(4 \mathbf { i } + 7 \mathbf { j } + 2 \mathbf { k }\) ) N and the normal reaction of the wire. The bead starts from rest at the point \(A\) with position vector \(( \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } ) \mathrm { m }\) and moves to the point \(B\) with position vector \(( 4 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } ) \mathrm { m }\).

Find the speed of the bead when it reaches \(B\).
(4)

2. At time \(t\) seconds, the position vector of a particle \(P\) is \(\mathbf { r }\) metres, where \(\mathbf { r }\) satisfies the differential equation $$\frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm {~d} t ^ { 2 } } + 3 \frac { \mathrm {~d} \mathbf { r } } { \mathrm {~d} t } = \mathbf { 0 }$$ When \(t = 0\), the velocity of \(P\) is \(( 8 \mathbf { i } - 12 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
Find the velocity of \(P\) when \(t = \frac { 2 } { 3 } \ln 2\).
(7)

5. Two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) act on a rigid body, where \(\mathbf { F } _ { 1 } = ( 3 \mathbf { i } + 4 \mathbf { j } - 6 \mathbf { k } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( 5 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } ) \mathrm { N }\).
The force \(\mathbf { F } _ { 1 }\) acts at the point with position vector \(( \mathbf { i } - 2 \mathbf { j } ) \mathrm { m }\), and the force \(\mathbf { F } _ { 2 }\) acts at the point with position vector ( \(3 \mathbf { i } - \mathbf { k }\) ) m. The two forces are equivalent to a single force \(\mathbf { F }\) acting at the point with position vector \(( \mathbf { i } - \mathbf { k } ) \mathrm { m }\), together with a couple \(\mathbf { G }\).

1. Find \(\mathbf { F }\).
2. Find the magnitude of \(\mathbf { G }\).
   (8)

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

A small bead of mass 0.5 kg is threaded on a smooth horizontal wire. The bead is initially at rest at the point with position vector \(( \mathbf { i } - 6 \mathbf { j } ) \mathrm { m }\). A constant horizontal force \(\mathbf { P } \mathrm { N }\) then acts on the bead causing it to move along the wire. The bead passes through the point with position vector ( \(7 \mathbf { i } - 14 \mathbf { j }\) ) m with speed \(2 \sqrt { 7 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that \(\mathbf { P }\) is parallel to ( \(6 \mathbf { i } + \mathbf { j }\) ), find \(\mathbf { P }\).
(6)

2. The velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) of a particle \(P\) at time \(t\) seconds satisfies the vector differential equation $$\frac { \mathrm { d } \mathbf { v } } { \mathrm {~d} t } + 4 \mathbf { v } = \mathbf { 0 }$$ The position vector of \(P\) at time \(t\) seconds is \(\mathbf { r }\) metres.
Given that at \(t = 0 , \mathbf { r } = ( \mathbf { i } - \mathbf { j } )\) and \(\mathbf { v } = ( - 8 \mathbf { i } + 4 \mathbf { j } )\), find \(\mathbf { r }\) at time \(t\) seconds.

3. A system of forces consists of two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) acting on a rigid body. \(\mathbf { F } _ { 1 } = ( - 2 \mathbf { i } + \mathbf { j } - \mathbf { k } ) \mathrm { N }\) and acts at the point with position vector \(\mathbf { r } _ { 1 } = ( \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { m }\). \(\mathbf { F } _ { 2 } = ( 3 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } ) \mathrm { N }\) and acts at the point with position vector \(\mathbf { r } _ { 2 } = ( 4 \mathbf { i } - \mathbf { j } - 2 \mathbf { k } ) \mathrm { m }\).
Given that the system is equivalent to a single force \(\mathbf { R } \mathrm { N }\), acting at the point with position vector \(( 5 \mathbf { i } + \mathbf { j } - \mathbf { k } ) \mathrm { m }\), together with a couple \(\mathbf { G N m }\), find

1. \(\mathbf { R }\),
2. the magnitude of \(\mathbf { G }\).
   (9)

2. A particle \(P\) moves so that its position vector, \(\mathbf { r }\) metres, at time \(t\) seconds, where \(0 \leqslant t < \frac { \pi } { 2 }\), satisfies the differential equation $$\frac { \mathrm { d } \mathbf { r } } { \mathrm {~d} t } - ( \tan t ) \mathbf { r } = ( \sin t ) \mathbf { i }$$ When \(t = 0 , \mathbf { r } = - \frac { 1 } { 2 } \mathbf { i }\).
Find \(\mathbf { r }\) in terms of \(t\).

2. [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane and \(\mathbf { k }\) is a unit vector vertically upwards.] A particle of mass 2 kg moves under the action of a constant gravitational force \(- 19.6 \mathbf { k } \mathrm {~N}\). The particle is subject to a resistive force \(- \mathbf { v }\) newtons, where \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) is the velocity of the particle at time \(t\) seconds.

1. By writing down an equation of motion of the particle, show that \(\mathbf { v }\) satisfies the differential equation $$\frac { \mathrm { d } \mathbf { v } } { \mathrm {~d} t } + 0.5 \mathbf { v } = - 9.8 \mathbf { k }$$ When \(t = 0 , \mathbf { v } = ( 4 \mathbf { i } - 6 \mathbf { j } + 11.6 \mathbf { k } )\)
2. Find \(\mathbf { v }\) when \(t = \ln 4\)