1.10d Vector operations: addition and scalar multiplication

A particle is acted upon by two forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\), given by \(\mathbf{F}_1 = (\mathbf{i} - 3\mathbf{j})\) N, \(\mathbf{F}_2 = (p\mathbf{i} + 2p\mathbf{j})\) N, where \(p\) is a positive constant.

1. Find the angle between \(\mathbf{F}_2\) and \(\mathbf{j}\). [2]

The resultant of \(\mathbf{F}_1\) and \(\mathbf{F}_2\) is \(\mathbf{R}\). Given that \(\mathbf{R}\) is parallel to \(\mathbf{i}\),

1. find the value of \(p\). [4]

[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors due east and due north respectively.] A hiker \(H\) is walking with constant velocity \((1.2\mathbf{i} - 0.9\mathbf{j})\) m s\(^{-1}\).

1. Find the speed of \(H\). [2]

\includegraphics{figure_3} A horizontal field \(OABC\) is rectangular with \(OA\) due east and \(OC\) due north, as shown in Figure 3. At twelve noon hiker \(H\) is at the point \(Y\) with position vector \(100\mathbf{j}\) m, relative to the fixed origin \(O\).

1. Write down the position vector of \(H\) at time \(t\) seconds after noon. [2]

At noon, another hiker \(K\) is at the point with position vector \((9\mathbf{i} + 46\mathbf{j})\) m. Hiker \(K\) is moving with constant velocity \((0.75\mathbf{i} + 1.8\mathbf{j})\) m s\(^{-1}\).

1. Show that, at time \(t\) seconds after noon, $$\overrightarrow{HK} = [(9 - 0.45t)\mathbf{i} + (2.7t - 54)\mathbf{j}] \text{ metres.}$$ [4]

Hence,

1. show that the two hikers meet and find the position vector of the point where they meet. [5]

[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors due east and due north respectively. Position vectors are given relative to a fixed origin \(O\).] Two ships \(P\) and \(Q\) are moving with constant velocities. Ship \(P\) moves with velocity \((2\mathbf{i} - 3\mathbf{j})\) km h\(^{-1}\) and ship \(Q\) moves with velocity \((3\mathbf{i} + 4\mathbf{j})\) km h\(^{-1}\).

1. Find, to the nearest degree, the bearing on which \(Q\) is moving. [2]

At 2 pm, ship \(P\) is at the point with position vector \((\mathbf{i} + \mathbf{j})\) km and ship \(Q\) is at the point with position vector \((-2\mathbf{j})\) km. At time \(t\) hours after 2 pm, the position vector of \(P\) is \(\mathbf{p}\) km and the position vector of \(Q\) is \(\mathbf{q}\) km.

1. Write down expressions, in terms of \(t\), for
   1. \(\mathbf{p}\),
   2. \(\mathbf{q}\),
   3. \(\overrightarrow{PQ}\). [5]
2. Find the time when
   1. \(Q\) is due north of \(P\),
   2. \(Q\) is north-west of \(P\). [4]

[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors due east and due north respectively. Position vectors are given with respect to a fixed origin \(O\).] A ship \(S\) is moving with constant velocity \((3\mathbf{i} + 3\mathbf{j})\) km h\(^{-1}\). At time \(t = 0\), the position vector of \(S\) is \((-4\mathbf{i} + 2\mathbf{j})\) km.

1. Find the position vector of \(S\) at time \(t\) hours. [2]

A ship \(T\) is moving with constant velocity \((-2\mathbf{i} + n\mathbf{j})\) km h\(^{-1}\). At time \(t = 0\), the position vector of \(T\) is \((6\mathbf{i} + \mathbf{j})\) km. The two ships meet at the point \(P\).

1. Find the value of \(n\). [5]
2. Find the distance \(OP\). [4]

[In this question, the horizontal unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are directed due east and due north respectively.] The velocity, \(\mathbf{v} \text{ m s}^{-1}\), of a particle \(P\) at time \(t\) seconds is given by $$\mathbf{v} = (1 - 2t)\mathbf{i} + (3t - 3)\mathbf{j}$$

1. Find the speed of \(P\) when \(t = 0\) [3]
2. Find the bearing on which \(P\) is moving when \(t = 2\) [2]
3. Find the value of \(t\) when \(P\) is moving
   1. parallel to \(\mathbf{j}\),
   2. parallel to \((-\mathbf{i} - 3\mathbf{j})\). [6]

\includegraphics{figure_2} Two particles \(A\) and \(B\) have masses \(2m\) and \(3m\) respectively. The particles are attached to the ends of a light inextensible string. Particle \(A\) is held at rest on a smooth horizontal table. The string passes over a small smooth pulley which is fixed at the edge of the table. Particle \(B\) hangs at rest vertically below the pulley with the string taut, as shown in Figure 2. Particle \(A\) is released from rest. Assuming that \(A\) has not reached the pulley, find

1. the acceleration of \(B\), [5]
2. the tension in the string, [1]
3. the magnitude and direction of the force exerted on the pulley by the string. [4]

[In this question, the horizontal unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are directed due East and North respectively.] A coastguard station \(O\) monitors the movements of ships in a channel. At noon, the station's radar records two ships moving with constant speed. Ship \(A\) is at the point with position vector \((-5\mathbf{i} + 10\mathbf{j})\) km relative to \(O\) and has velocity \((2\mathbf{i} + 2\mathbf{j})\) km h\(^{-1}\). Ship \(B\) is at the point with position vector \((3\mathbf{i} + 4\mathbf{j})\) km and has velocity \((-2\mathbf{i} + 5\mathbf{j})\) km h\(^{-1}\).

1. Given that the two ships maintain these velocities, show that they collide. [6]

The coast guard radios ship \(A\) and orders it to reduce its speed to move with velocity \((\mathbf{i} + \mathbf{j})\) km h\(^{-1}\). Given that \(A\) obeys this order and maintains this new constant velocity,

1. find an expression for the vector \(\overrightarrow{AB}\) at time \(t\) hours after noon. [2]
2. find, to 3 significant figures, the distance between \(A\) and \(B\) at 1400 hours, [3]
3. Find the time at which \(B\) will be due north of \(A\). [2]

Two cars \(A\) and \(B\) are moving on straight horizontal roads with constant velocities. The velocity of \(A\) is \(20 \text{ m s}^{-1}\) due east, and the velocity of \(B\) is \((10\mathbf{i} + 10\mathbf{j}) \text{ m s}^{-1}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors directed due east and due north respectively. Initially \(A\) is at the fixed origin \(O\), and the position vector of \(B\) is \(300\mathbf{j}\) m relative to \(O\). At time \(t\) seconds, the position vectors of \(A\) and \(B\) are \(\mathbf{r}\) metres and \(\mathbf{s}\) metres respectively.

1. Find expressions for \(\mathbf{r}\) and \(\mathbf{s}\) in terms of \(t\). [3]
2. Hence write down an expression for \(\overrightarrow{AB}\) in terms of \(t\). [1]
3. Find the time when the bearing of \(B\) from \(A\) is \(045°\). [5]
4. Find the time when the cars are again 300 m apart. [6]

The point \(A\) has coordinates \((4, -3, 2)\). The line \(l_1\) passes through \(A\) and has equation \(\mathbf{r} = \begin{bmatrix} 4 \\ -3 \\ 2 \end{bmatrix} + \lambda \begin{bmatrix} 2 \\ 0 \\ 1 \end{bmatrix}\). The line \(l_2\) has equation \(\mathbf{r} = \begin{bmatrix} -1 \\ 3 \\ 4 \end{bmatrix} + \mu \begin{bmatrix} 1 \\ -2 \\ -1 \end{bmatrix}\). The point \(B\) lies on \(l_2\) where \(\mu = 2\).

1. Find the vector \(\overrightarrow{AB}\). [3 marks]
2. 1. Show that the lines \(l_1\) and \(l_2\) intersect. [4 marks]
   2. The lines \(l_1\) and \(l_2\) intersect at the point \(P\). Find the coordinates of \(P\). [1 mark]
3. The point \(C\) lies on a line which is parallel to \(l_1\) and which passes through the point \(B\). The points \(A\), \(B\), \(C\) and \(P\) are the vertices of a parallelogram. Find the coordinates of the two possible positions of the point \(C\). [4 marks]

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 \(\mathbf{r} = \begin{bmatrix} -1 \\ 5 \\ -2 \end{bmatrix} + \lambda \begin{bmatrix} 3 \\ -5 \\ 1 \end{bmatrix}\).

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. [5 marks]
2. Show that the lines \(l_1\) and \(l_2\) intersect and find the coordinates of the point of intersection. [5 marks]
3. The points \(C\) and \(D\) lie on line \(l_2\) such that \(ACBD\) is a parallelogram. \includegraphics{figure_6} The length of \(AB\) is three times the length of \(CD\). Find the coordinates of the points \(C\) and \(D\). [5 marks]

Relative to a fixed origin \(O\), the point \(A\) has position vector \(3\mathbf{i} + 2\mathbf{j} - \mathbf{k}\), the point \(B\) has position vector \(5\mathbf{i} + \mathbf{j} + \mathbf{k}\), and the point \(C\) has position vector \(7\mathbf{i} - \mathbf{j}\).

1. Find the cosine of angle \(ABC\). [4]
2. Find the exact value of the area of triangle \(ABC\). [3]

The point \(D\) has position vector \(7\mathbf{i} + 3\mathbf{k}\).

1. Show that \(AC\) is perpendicular to \(CD\). [2]
2. Find the ratio \(AD:DB\). [2]

The position vectors of three points \(A\), \(B\) and \(C\) relative to an origin \(O\) are given respectively by $$\overrightarrow{OA} = 7\mathbf{i} + 3\mathbf{j} - 3\mathbf{k},$$ $$\overrightarrow{OB} = 4\mathbf{i} + 2\mathbf{j} - 4\mathbf{k}$$ and $$\overrightarrow{OC} = 5\mathbf{i} + 4\mathbf{j} - 5\mathbf{k}.$$

1. Find the angle between \(AB\) and \(AC\). [6]
2. Find the area of triangle \(ABC\). [2]

The points A, B and C have coordinates \(A(3, 2, -1)\), \(B(-1, 1, 2)\) and \(C(10, 5, -5)\), relative to the origin O. Show that \(\overrightarrow{OC}\) can be written in the form \(\lambda\overrightarrow{OA} + \mu\overrightarrow{OB}\), where \(\lambda\) and \(\mu\) are to be determined. What can you deduce about the points O, A, B and C from the fact that \(\overrightarrow{OC}\) can be expressed as a combination of \(\overrightarrow{OA}\) and \(\overrightarrow{OB}\)? [6]

A bee flies in a straight line from \(A\) to \(B\), where \(\overrightarrow{AB} = (3\mathbf{i} - 12\mathbf{j})\) m, in 5 seconds at a constant speed. Find

1. the straight-line distance \(AB\), [2 marks]
2. the speed of the bee, [2 marks]
3. the velocity vector of the bee. [2 marks]

Two trains \(S\) and \(T\) are moving with constant speeds on straight tracks which intersect at the point \(O\). At 9.00 a.m. \(S\) has position vector \((-10\mathbf{i} + 24\mathbf{j})\) km and \(T\) has position vector \(25\mathbf{j}\) km relative to \(O\), where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors in the directions due east and due north respectively. \(S\) is moving with speed 52 km h\(^{-1}\) and \(T\) is moving with speed 50 km h\(^{-1}\), both towards \(O\).

1. Show that the velocity vector of \(S\) is \((20\mathbf{i} - 48\mathbf{j})\) km h\(^{-1}\) and find the velocity vector of \(T\). \hfill [5 marks]
2. Find expressions for the position vectors of \(S\) and \(T\) at time \(t\) minutes after 9.00 a.m. \hfill [5 marks]
3. Show that the bearing of \(T\) from \(S\) remains constant during the motion, and find this bearing. \hfill [5 marks]
4. Show that if the trains continue at the given speeds they will collide. \hfill [2 marks]

Two forces each of magnitude \(4\text{ N}\) have a resultant of magnitude \(6\text{ N}\).

1. Calculate the angle between the two \(4\text{ N}\) forces. [4]

The two given forces of magnitude \(4\text{ N}\) act on a particle of mass \(m\text{ kg}\) which remains at rest on a smooth horizontal surface. The surface exerts a force of magnitude \(3\text{ N}\) on the particle.

1. Find \(m\), and give the acute angle between the surface and one of the \(4\text{ N}\) forces. [3]

The force acting on a particle of mass 1.5 kg is given by the vector \(\begin{pmatrix} 6 \\ 9 \end{pmatrix}\) N.

1. Give the acceleration of the particle as a vector. [2]
2. Calculate the angle that the acceleration makes with the direction \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\). [2]
3. At a certain point of its motion, the particle has a velocity of \(\begin{pmatrix} -2 \\ 3 \end{pmatrix}\) m s\(^{-1}\). Calculate the displacement of the particle over the next two seconds. [3]

Force \(\mathbf{F}\) is \(\begin{pmatrix} 4 \\ 1 \\ 2 \end{pmatrix}\) N and force \(\mathbf{G}\) is \(\begin{pmatrix} -6 \\ 2 \\ 4 \end{pmatrix}\) N.

1. Find the resultant of \(\mathbf{F}\) and \(\mathbf{G}\) and calculate its magnitude. [4]
2. Forces \(\mathbf{F}\), \(2\mathbf{G}\) and \(\mathbf{H}\) act on a particle which is in equilibrium. Find \(\mathbf{H}\). [3]

Particle \(A\) has velocity \((8\mathbf{i} - 3\mathbf{j})\) ms\(^{-1}\) and particle \(B\) has velocity \((15\mathbf{i} - 8\mathbf{j})\) ms\(^{-1}\) where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular, horizontal unit vectors.

1. Find the speed of \(B\). [2 marks]
2. Find the velocity of \(B\) relative to \(A\). [2 marks]
3. Find the acute angle between the relative velocity found in part (b) and the vector \(\mathbf{i}\), giving your answer in degrees correct to 1 decimal place. [2 marks]

A constant force, \(\mathbf{F}\), acts on a particle, \(P\), of mass 5 kg causing its velocity to change from \((-2\mathbf{i} + \mathbf{j})\) m s\(^{-1}\) to \((4\mathbf{i} - 7\mathbf{j})\) m s\(^{-1}\) in 2 seconds.

1. Find, in the form \(a\mathbf{i} + b\mathbf{j}\), the acceleration of \(P\). [2 marks]
2. Show that the magnitude of \(\mathbf{F}\) is 25 N and find, to the nearest degree, the acute angle between the line of action of \(\mathbf{F}\) and the vector \(\mathbf{j}\). [5 marks]

Two ramblers, Alison and Bill, are out walking. At midday, Alison is at the fixed origin \(O\), and Bill is at the point with position vector \((-5\mathbf{i} + 12\mathbf{j})\) km relative to \(O\), where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular, horizontal unit vectors. They are both walking with constant velocity – Alison at \((2\mathbf{i} + 5\mathbf{j})\) km h\(^{-1}\), and Bill at a speed of \(2\sqrt{10}\) km h\(^{-1}\) in a direction parallel to the vector \((3\mathbf{i} + \mathbf{j})\).

1. Find the distance between the two ramblers at midday. [2 marks]
2. Show that the velocity of Bill is \((6\mathbf{i} + 2\mathbf{j})\) km h\(^{-1}\). [3 marks]
3. Show that, at time \(t\) hours after midday, the position vector of Bill relative to Alison is $$[(4t - 5)\mathbf{i} + (12 - 3t)\mathbf{j}] \text{ km.}$$ [5 marks]
4. Show that the distance, \(d\) km, between the two ramblers is given by $$d^2 = 25t^2 - 112t + 169.$$ [2 marks]
5. Using your answer to part \((d)\), find the length of time to the nearest minute for which the distance between the Alison and Bill is less than 11 km. [5 marks]

During a cricket match, the batsman hits the ball and begins running with constant velocity \(4\mathbf{i}\) m s\(^{-1}\) to try and score a run. When the batsman is at the fixed origin \(O\), the ball is thrown by a member of the opposing team with velocity \((^-8\mathbf{i} + 24\mathbf{j})\) m s\(^{-1}\) from the point with position vector \((30\mathbf{i} - 60\mathbf{j})\) m, where \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal perpendicular unit vectors. At time \(t\) seconds after the ball is thrown, the position vectors of the batsman and the ball are \(\mathbf{r}\) metres and \(\mathbf{s}\) metres respectively. In a model of the situation, the ball is assumed to travel horizontally and air resistance is considered to be negligible.

1. Find expressions for \(\mathbf{r}\) and \(\mathbf{s}\) in terms of \(t\). [3 marks]
2. Show that the ball hits the batsman and find the position vector of the batsman when this occurs. [5 marks]
3. Write down two reasons why the assumptions used in these calculations are unlikely to provide a realistic model. [2 marks]

\(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors in a horizontal plane. A body of mass 1 kg moves under the action of a constant force \((4\mathbf{i} + 5\mathbf{j})\) N. The body moves from the point \(P\) with position vector \((-3\mathbf{i} - 15\mathbf{j})\) m to the point \(Q\) with position vector \(9\mathbf{i}\) m.

1. Find the work done by the force in moving the body from \(P\) to \(Q\). [5 marks]
2. Given that the body started from rest at \(P\), find its speed when it is at \(Q\). [5 marks]

In this question use \(\cos 30° = \sin 60° = \frac{\sqrt{3}}{2}\). A smooth spherical ball, \(A\), is moving with speed \(u\) in a straight line on a smooth horizontal table when it hits an identical ball, \(B\), which is at rest on the table. Just before the collision, the direction of motion of \(A\) is parallel to a fixed smooth vertical wall. At the instant of collision, the line of centres of \(A\) and \(B\) makes an angle of \(60°\) with the wall, as shown in the diagram. \includegraphics{figure_6} The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. Show that the speed of \(B\) immediately after the collision is \(\frac{1}{4}u(1 + e)\) and find, in terms of \(u\) and \(e\), the components of the velocity of \(A\), parallel and perpendicular to the line of centres, immediately after the collision. [7 marks]
2. Subsequently, \(B\) collides with the wall. After colliding with the wall, the direction of motion of \(B\) is parallel to the direction of motion of \(A\) after its collision with \(B\). Show that the coefficient of restitution between \(B\) and the wall is \(\frac{1 + e}{7 - e}\). [7 marks]

A quad-bike, a truck and a car are moving on a large, open, horizontal surface in a desert plain. Relative to the quad-bike, which is travelling due west at its maximum speed of \(10 \text{ m s}^{-1}\), the truck is moving on a bearing of \(340°\). Relative to the car, which is travelling due east at a speed of \(15 \text{ m s}^{-1}\), the truck is moving on a bearing of \(300°\).

1. Show that the speed of the truck is approximately \(24.7 \text{ m s}^{-1}\) and that it is moving on a bearing of \(318°\), correct to the nearest degree. [8 marks]
2. At the instant when the truck is at a distance of \(400\) metres from the quad-bike, the bearing of the truck from the quad-bike is \(060°\). The truck continues to move with the same velocity as in part (a). The quad-bike continues to move at a speed of \(10 \text{ m s}^{-1}\). Find the bearing, to the nearest degree, on which the quad-bike should travel in order to approach the truck as closely as possible. [5 marks]