Geometry and trigonometry

A fishing boat leaves a mooring and moves with constant velocity. Relative to a harbour at the origin O, its position vector \(t\) hours after 08:00 is
\(\mathbf{r} = \begin{pmatrix}2\\-1\end{pmatrix} + t\begin{pmatrix}3\\4\end{pmatrix}\)
where distances are in kilometres, the first component is east of O and the second is north of O.

A survey drone takes off from the roof of a building. Relative to a point O at ground level directly below the take-off point, the drone's position \(t\) seconds after take-off is
\(\mathbf{r} = \begin{pmatrix}0\\0\\20\end{pmatrix} + t\begin{pmatrix}4\\-3\\1.5\end{pmatrix}\)
where distances are in metres and the third component is the height above the ground.

A road inspector is responsible for the roads joining four villages, A, B, C and D. There are direct roads between A and B, A and C, B and C, B and D, and C and D. There are no other roads. The road network is modelled by a graph G whose vertices are the villages and whose edges are the roads.

A college is laying fibre-optic cable to connect six buildings, A, B, C, D, E and F. The cable can only be laid along the following routes. Each length is given in metres.
AB 120, AC 90, AD 150, BC 70, BE 110, CD 100, CE 140, DE 80, DF 130, EF 60.
Every building must be connected to the network, and the total length of cable must be as small as possible.

A rescue boat leaves a harbour buoy. Its position vector, in kilometres relative to a lighthouse at the origin O, is given by
\(\mathbf{r} = \begin{pmatrix} 2 \\ -1 \end{pmatrix} + t\begin{pmatrix} 3 \\ 4 \end{pmatrix}\)
where t is the time in hours after leaving the buoy. The unit vectors point east and north respectively.

A survey drone takes off from a point with position vector \(\begin{pmatrix} 10 \\ 5 \\ 0 \end{pmatrix}\) m, measured from a ground station. It flies with constant velocity \(\begin{pmatrix} 4 \\ -2 \\ 1.5 \end{pmatrix}\) m s⁻¹, where the third component is vertical.

A fishing boat leaves a harbour. Relative to a lighthouse at the origin O, the position vector of the boat t hours after leaving the harbour is
\(\boldsymbol{r} = \begin{pmatrix}3\\-2\end{pmatrix} + t\begin{pmatrix}4\\3\end{pmatrix}\)
where distances are in kilometres, \(\boldsymbol{i}\) points east and \(\boldsymbol{j}\) points north.

A plane is pointed due north and flies at 300 km h⁻¹ relative to the air. A wind blows with velocity \(\begin{pmatrix}40\\-30\end{pmatrix}\) km h⁻¹, where \(\boldsymbol{i}\) points east and \(\boldsymbol{j}\) points north.

A delivery drone moves in a straight line after take-off. Relative to an origin at the base of its charging station, its position vector t seconds after take-off is
r = (3, −2, 1) + t(4, 5, 2),
where the components are in metres, x is east, y is north and z is vertically upwards. Ground level is z = 0.

Four villages, A, B, C and D, are linked by roads. There are direct roads between A and B, A and C, B and C, B and D, and C and D. There are no other direct roads. The road network is modelled by a graph G whose vertices are the villages and whose edges are the roads.

A university plans to connect six buildings, P, Q, R, S, T and U, with fibre-optic cable. Cable can only be laid along the routes listed below. The lengths are in metres.
PQ 120, PR 90, QR 70, QS 150, RS 110, RT 130, ST 80, SU 100, TU 95, QU 200
Every building must be connected to the network, and the total length of cable must be as small as possible.

A ferry leaves a harbour, taken as the origin, and travels at a constant speed of 12 km h⁻¹ on a bearing of 060°. Position vectors are given as (east component, north component) in kilometres, and t is the time in hours after the ferry leaves the harbour. A lighthouse is located at (20, 4).

A survey drone flies in a straight line from A(2, −1, 5) to B(8, 3, 17), where coordinates are in metres relative to a base station.

Two tugboats pull a barge. The forces they exert are \(\mathbf{F}_1=\begin{pmatrix}8\\6\end{pmatrix}\) kN and \(\mathbf{F}_2=\begin{pmatrix}5\\-12\end{pmatrix}\) kN.

Four islands A, B, C and D are linked by ferry routes: A–B, A–C, B–C, B–D and C–D. Each route can be travelled in either direction.

Three villages are at A(0, 0), B(6, 0) and C(0, 8) on a map (1 unit = 1 km). A wind turbine is to be built at the point that is the same distance from all three villages.

In a retail warehouse, the directions of two conveyor belts are given by the vectors a = 3i − 2j + 4k and b = i + 5j − 2k.

A retailer's sign shows a triangle with vertices A(2, 1), B(6, 1) and C(6, 4). The triangle is transformed by the matrix \(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\).

The contacts between four patients A, B, C and D on a ward are modelled by a graph. The edges are AB, AC, BC and CD.

Five solar arrays P, Q, R, S and T are to be connected by cable. The possible cable lengths in km are:
PQ = 4, PR = 7, PS = 9, QR = 3, QS = 8, QT = 10, RS = 5, RT = 6, ST = 2.

A light aircraft leaves airfield A and flies 240 km on a bearing of 065° to airfield B. It then flies 180 km on a bearing of 150° to airfield C.

A triangular field PQR has PQ = 420 m, PR = 350 m and angle QPR = 72°.

Two aircraft leave the same airport at the same time. Relative to the airport, with i pointing east, j north and k vertically upwards, their velocities are v₁ = (200, 150, 20) km h⁻¹ and v₂ = (180, −60, 15) km h⁻¹.

An airline operates direct routes between five airports A, B, C, D and E. The adjacency matrix below shows the number of direct routes between each pair of airports, with rows and columns in the order A, B, C, D, E.
0  1  2  0  0
1  0  1  1  0
2  1  0  1  1
0  1  1  0  2
0  0  1  2  0

Positions on a mountain are given relative to base camp, in metres, using i for east, j for north and k for vertically upwards. Camp 1 is at (120, 350, 640) and Camp 2 is at (−80, 510, 930).

A bakery designs a decoration stencil in the shape of triangle PQR, where P = (1, 1), Q = (4, 1) and R = (1, 3), measured in centimetres. A printing process transforms every point (x, y) using the matrix \(M=\begin{pmatrix}3&1\\-1&2\end{pmatrix}\).

Mountaineers leave a hut H and walk 5.2 km on a bearing of 072° to a point P. They then walk 3.8 km on a bearing of 150° to a point Q. Give lengths to 3 significant figures and bearings to the nearest degree.

Two lighting beams from a rig at the origin O on a theatre ceiling point in the directions \(\mathbf{u}=\begin{pmatrix}3\\-2\\6\end{pmatrix}\) and \(\mathbf{v}=\begin{pmatrix}2\\4\\-1\end{pmatrix}\).

A shipping company operates direct routes between five ports A, B, C, D and E. The direct routes are AB, AC, BC, BD, CD and DE; each can be sailed in either direction.

A port authority plans to connect six container terminals A to F by rail links so that every terminal can be reached from every other. The possible links and their construction costs, in millions of euros, are:
AB 12, AC 9, BC 7, BD 15, CD 11, CE 14, DE 8, DF 10, EF 6.

A volcano is modelled as a cone with a circular base of radius 1.8 km and a vertical height of 0.9 km.

A sprinter in a training session is tracked on a flat field. Her position vector relative to a marker at the origin is \(\mathbf{r} = \begin{pmatrix}20\\5\end{pmatrix} + t\begin{pmatrix}3.5\\-1.2\end{pmatrix}\), where distances are in metres and t is the time in seconds, \(t \geq 0\).

Two thrusters on a space probe produce forces (in newtons) represented by the vectors \(\mathbf{F}_1 = \begin{pmatrix}2\\3\\-1\end{pmatrix}\) and \(\mathbf{F}_2 = \begin{pmatrix}5\\-1\\4\end{pmatrix}\).

On a coaching board, a player's position (x, y) is transformed by a rotation of 90° anticlockwise about the origin, followed by an enlargement with scale factor 2 centred at the origin.

A sports hall is a cuboid 40 m long, 25 m wide and 12 m high. A cable is stretched in a straight line from floor corner A to the opposite roof corner G (diagonally opposite through the hall).

A hospital has five departments: Admissions (A), Biochemistry (B), Cardiology (C), Dispensary (D) and Exit (E). Corridors join the following pairs: AB, AC, BC, BD, CD and DE. Each corridor is a single edge of an undirected graph and the adjacency matrix M uses the order A, B, C, D, E.

A hospital logo is drawn as triangle PQR with P(2, 1), Q(5, 1) and R(5, 4). The matrix \(\mathbf{M}=\begin{pmatrix}0 & -1\\ 1 & 0\end{pmatrix}\) transforms the logo.

An air ambulance is based at H. A hospital A is 18 km from H on a bearing of 060°, and an accident site B is 25 km from H on a bearing of 140°.

A shelter over a row of hives is a pyramid with a horizontal square base ABCD of side 6 m. The apex V is 4 m vertically above the centre O of the base.

Two Formula 1 cars are on different straights. At a particular instant, car P has velocity \(\begin{pmatrix}k\\60\end{pmatrix}\) m s⁻¹ and car Q has velocity \(\begin{pmatrix}90\\-120\end{pmatrix}\) m s⁻¹, where the components are east and north. The two velocities are perpendicular.

Five wards P, Q, R, S and T of a hospital are joined by corridors. The adjacency matrix M, with rows and columns in the order P, Q, R, S, T, is
(0 1 0 1 0)
(1 0 1 0 1)
(0 1 0 1 1)
(1 0 1 0 1)
(0 1 1 1 0).
Each corridor joins two wards and is shown by 1 in the matrix.

An ambulance leaves base A and drives 12 km on a bearing of 075° to a hospital B. It then drives 9 km on a bearing of 160° to a clinic C.

A hospital ward is a cuboid of length 12 m, width 8 m and height 3.5 m. A sensor is fixed at corner A on the floor, and a second sensor is fixed at corner B on the ceiling, diagonally opposite A.

Two straight railway lines leave a junction J. Station A is 12.4 km from J on a bearing of 072°. Station B is 9.8 km from J on a bearing of 145°.

A university observatory building is a vertical cylinder of radius 6 m and height 4 m, topped by a hemispherical dome of radius 6 m that sits exactly on the cylinder's top face.

In a sports science study, the position of a marker on a pitch is recorded relative to the centre spot, in metres. A marker is at \(P(4,2)\). The matrix \(\mathbf{R}=\begin{pmatrix}0&-1\\1&0\end{pmatrix}\) represents an anticlockwise rotation of 90° about the centre spot.

A cyclist in a training session has position vector \(\mathbf{r}=\begin{pmatrix}30\\10\end{pmatrix}+t\begin{pmatrix}4.5\\6.0\end{pmatrix}\) metres relative to a coach, where \(t\) is the time in seconds after the start. The unit vectors point east and north respectively.

A swimmer crosses a straight river that is 60 m wide. In still water she swims at 1.2 m s⁻¹. She aims directly north, perpendicular to the banks. The river flows east at 0.5 m s⁻¹. Use unit vectors i (east) and j (north).

Five libraries A, B, C, D and E are connected by direct roads. The adjacency matrix M for the network is shown, with rows and columns in the order A, B, C, D, E.
A: (0 1 1 0 0)
B: (1 0 1 1 0)
C: (1 1 0 1 0)
D: (0 1 1 0 1)
E: (0 0 0 1 0)

A wind farm has three turbines A, B and C on flat ground. Turbine B is 6.4 km from A on a bearing of 070°. Turbine C is 4.8 km from A on a bearing of 150°.

A swimming pool is 25 m long and 12 m wide. The floor slopes uniformly from a depth of 1.2 m at the shallow end to 3.8 m at the deep end.

Two thrusters on a space probe produce forces (in kN) F₁ = 2i − j + 3k and F₂ = 4i + 2j − k.

A rover explores a lunar plain. Its position relative to the landing module is r(t) = (t³ − 2t)i + (5t − t²)j, where distances are in metres and t is the time in seconds, t ≥ 0.

A satellite design is drawn on a coordinate grid. A rotation of 90° anticlockwise about the origin is represented by the matrix R = \(\begin{pmatrix}0 & -1\\1 & 0\end{pmatrix}\), and a reflection in the x-axis is represented by S = \(\begin{pmatrix}1 & 0\\0 & -1\end{pmatrix}\).

A drone inspects a suspension bridge. It leaves tower T and flies 340 m on a bearing of 065° to a point P. It then flies 280 m on a bearing of 150° to a point Q.

An underwater inspection robot at a fish farm moves from point P to point Q. Relative to a fixed buoy, the position vectors are \(\overrightarrow{OP}=\begin{pmatrix}2\\-1\\-4\end{pmatrix}\) m and \(\overrightarrow{OQ}=\begin{pmatrix}8\\3\\-16\end{pmatrix}\) m, where the third component is the height relative to the surface (negative below the surface).

Two segments of a robotic arm are modelled by the vectors \(\mathbf{a}=\begin{pmatrix}2\\k\\-3\end{pmatrix}\) and \(\mathbf{b}=\begin{pmatrix}4\\-1\\2\end{pmatrix}\), where \(k\) is a constant.

A triangular fabric print has vertices A(1, 1), B(4, 1) and C(1, 3), measured in cm. A design software applies the transformation represented by the matrix \(\mathbf{T}=\begin{pmatrix}2&1\\0&3\end{pmatrix}\).

A robotic arm moves its gripper in a straight line from point A(2, −1, 3) to point B(8, 3, 5). Coordinates are in metres relative to the base of the arm.

A warehouse robot has constant acceleration \(\mathbf{a}=\begin{pmatrix}2\\-1\end{pmatrix}\) m s⁻². At time \(t=0\) it is at position \(\begin{pmatrix}5\\-2\end{pmatrix}\) m with velocity \(\begin{pmatrix}1\\3\end{pmatrix}\) m s⁻¹.

Five charging stations A, B, C, D and E in a warehouse are linked by tracks used by robots. The tracks are AB, AC, BC, BD, CE and DE. Each track can be used in either direction.

Three boutiques are located at A(2, 1), B(8, 1) and C(5, 7), with coordinates in kilometres. The Voronoi diagram for these sites has one vertex. The perpendicular bisector of AB is the line \(x=5\).

A research ship monitors a volcanic island. Relative to a harbour at O, the position vector of the ship t hours after 09:00 is r = (4, 10) + t(6, −8), where components are in kilometres, the first being east and the second north.

Two spotlights in a theatre shine along beams with direction vectors a = 3i − 2j + 6k and b = 2i + 3j − k.

A cargo ship leaves port P and sails 12 km on a bearing of 065° to a buoy B. It then sails 9 km on a bearing of 140° to a point C.

From a point A on level ground, the angle of elevation of the summit S of a volcano is 12°. An observer walks 600 m directly towards the volcano, along a straight horizontal line, to a point B. From B the angle of elevation of S is 17°. A, B and the point directly below S are in the same vertical plane.

A robotic arm has two segments. Relative to the shoulder joint at the origin O (units in decimetres), the first segment is represented by the vector \(\mathbf{u}=\begin{pmatrix}2\\1\\2\end{pmatrix}\) and the second segment, starting at the end of the first, by \(\mathbf{v}=\begin{pmatrix}-1\\2\\2\end{pmatrix}\).

Four fishing ports P, Q, R and S are linked by direct sea routes: P–Q, P–R, Q–R, Q–S and R–S. There is no direct route between P and S. The routes are two-way.

A fashion print repeats a motif of area 12 cm². A computer transforms the motif using the matrix \(\mathbf{M}=\begin{pmatrix}2&1\\-1&3\end{pmatrix}\).

A cuboid fish crate has a rectangular base ABCD with AB = 60 cm and BC = 40 cm, and height 30 cm. A thermometer probe lies along the line from base corner A to the opposite top corner G, directly above C.

A crop-spraying aircraft takes off from a point O on a farm. Relative to O, its position vector in kilometres at time t hours after take-off is r = \(\begin{pmatrix}12\\-8\end{pmatrix}\) + t\(\begin{pmatrix}150\\90\end{pmatrix}\).

A farmer has five sheds, A, B, C, D and E, joined by tracks. There is exactly one track between each of these pairs of sheds: AB, AC, BC, BD, CD, CE and DE. There are no other tracks.

A music app animates a note head on a screen using transformations of the plane. Transformation A is a rotation of 90° anticlockwise about the origin. Transformation B is an enlargement with scale factor 3 and centre the origin. The transformation A is applied first, followed by B.

A farmer inspects five fields, A to E, by walking along the tracks between them. The lengths of the tracks in kilometres are: AB 4, AC 6, BC 5, BD 9, CD 3, CE 8 and DE 5. The farmer wants to walk along every track at least once, starting and finishing at the same field, with the shortest total distance.

A cargo ship is tracked from a harbour at the origin O. Distances are in kilometres, with x measured east and y north. At time t = 0 hours the ship is at (−20, 30) and it moves with constant velocity (9, −12) km h⁻¹.

In a theatre, two spotlights hang from cables attached to a rigging point R. Taking R as the origin, the spotlights are at points A and B, where RA = (4, −3, −12) and RB = (−2, 6, −9). All coordinates are in metres.

A volcano is modelled as a right circular cone with base radius 1.7 km and vertical height 0.85 km.

An aircraft approaches a small airport. Relative to the airport, which is at the origin O, the position vector of the aircraft, in km, is given by \(\mathbf{r}=\begin{pmatrix}-60\\-80\end{pmatrix}+t\begin{pmatrix}90\\120\end{pmatrix}\), where t is the time in hours after 14:00.

Five airports A, B, C, D and E are linked by direct two-way flights on these routes only: AB, AC, AE, BC, BD, CD, CE and DE.

A designer scales and turns a triangular stage backdrop. The transformation is an enlargement, scale factor 1.5 and centre the origin, followed by an anticlockwise rotation of 90° about the origin.

A tagged dolphin is at position (2, 5) km relative to a research buoy at 08:00. It then swims with constant velocity (3, −4) km h⁻¹. Let t be the time in hours after 08:00.

Four fish-processing depots A, B, C and D are linked by direct roads: AB, AC, BC, BD and CD. Each road can be travelled in both directions. Let M be the adjacency matrix for the depots, with rows and columns in the order A, B, C, D.

A reef map has sampling points P(2, 1), Q(5, 1) and R(2, 4). The map is rotated 90° anticlockwise about the origin, and then enlarged by scale factor 1.5 with centre the origin.

Two support cables on a wind turbine tower have direction vectors u = (k, 3, −2) and v = (2, k, 5), where k is a constant.

Five retail stores A to E are linked by delivery roads. The adjacency matrix M, with rows and columns in the order A, B, C, D, E, is
\(\begin{pmatrix}0&1&1&0&1\\1&0&0&1&0\\1&0&0&1&1\\0&1&1&0&0\\1&0&1&0&0\end{pmatrix}\)

A solar array on a field plan is triangle PQR with P(1, 1), Q(4, 1) and R(1, 3). A horizontal shear is applied using the matrix S = \(\begin{pmatrix}1&2\\0&1\end{pmatrix}\).

A rectangular storage warehouse is 30 m long, 18 m wide and 9 m high. A cable runs from one floor corner F to the opposite roof corner G.

Two camps on a mountain are described using position vectors relative to a base point O, where the x-axis points east, the y-axis points north and the z-axis points vertically up. All distances are in km. Camp C₁ is at (1.2, 0.8, 3.4) and camp C₂ is at (2.9, 2.1, 4.6).

A telecoms company connects five relay towers A, B, C, D and E with direct cable links. The links are AB, AC, BC, BD, CE and DE. Each link can carry signals in both directions.

A park ranger must inspect every one of seven mountain trails. The trails join five huts A, B, C, D and E. The trail lengths in km are: AB 5, AC 7, BC 4, BD 8, CD 6, CE 9 and DE 3.

From a base camp B, a summit S lies on a bearing of 040° at a distance of 5.2 km. A mountain hut H lies on a bearing of 115° from B at a distance of 3.8 km.

A mountaineering team sets up base camp C and plans to climb to the summit S. Relative to a fixed origin, with i pointing east, j north and k vertically upwards (all distances in km), the position vectors are
r_C = 1.2i + 0.8j + 1.5k and r_S = 3.6i + 2.0j + 3.9k.

Four telecom towers A, B, C and D are linked by cables. The number of direct cables between each pair of towers is given by the adjacency matrix below (rows and columns in the order A, B, C, D).
A: 0 1 1 0
B: 1 0 1 1
C: 1 1 0 1
D: 0 1 1 0

On a plan of a telecom site, each mast is a point (x, y). The plan is first rotated 90° anticlockwise about the origin and then enlarged by scale factor 2 with centre the origin.

In a beekeeping apiary, three hives A, B and C stand on a hillside. Relative to the entrance gate, their position vectors, in metres, are \(\mathbf{a}=\begin{pmatrix}2\\3\\1\end{pmatrix}\), \(\mathbf{b}=\begin{pmatrix}8\\-1\\4\end{pmatrix}\) and \(\mathbf{c}=\begin{pmatrix}-1\\5\\2\end{pmatrix}\).

A Formula 1 car is driven along a straight practice stretch. Relative to a marker post, its position vector at time \(t\) seconds is \(\mathbf{r}=\begin{pmatrix}40t+3t^2\\25t-t^2\end{pmatrix}\) metres. Give answers to 3 significant figures where they are not exact.

A hospital has five wards A, B, C, D and E joined by corridors. The adjacency matrix \(\mathbf{M}\) below has rows and columns in the order A, B, C, D, E, and each corridor can be walked in either direction.
\(\mathbf{M}=\begin{pmatrix}0&1&1&0&0\\1&0&1&1&0\\1&1&0&1&0\\0&1&1&0&1\\0&0&0&1&0\end{pmatrix}\)

A light aircraft leaves a holding point and flies at constant velocity. Its position vector relative to a small airfield at the origin O is r = (−45, 30) + t(120, −90), where components are in kilometres (east, north) and t is the time in hours after leaving the holding point.

A sound engineer sits at the mixing desk, which is taken as the origin O on a plan of a concert hall. The positions of the left and right stage speakers, in metres relative to O (east, north), are L(−3.0, 5.0) and R(4.0, 6.0).

Two crop-spraying drones fly at constant velocity. Relative to the farm gate, with components in kilometres (east, north) and t in hours after 08:00, their positions are
drone 1: r₁ = (10, 5) + t(3, 2)
drone 2: r₂ = (30, −5) + t(−4, 9).

A space probe is moving through a region of space in a straight line at constant velocity. Relative to a mission origin O, its position vector at time \(t=0\) is \(\mathbf{r}_0 = \begin{pmatrix}120\\-45\\300\end{pmatrix}\) km and its velocity is \(\mathbf{v} = \begin{pmatrix}2.4\\-1.8\\3.0\end{pmatrix}\) km s⁻¹.

Two thrusters on a probe produce forces in the directions of the vectors \(\mathbf{u}=\begin{pmatrix}p\\2\\-3\end{pmatrix}\) and \(\mathbf{w}=\begin{pmatrix}4\\p\\1\end{pmatrix}\), where \(p\) is a constant.

A cargo ship carrying green coffee beans leaves port P and sails 48 km on a bearing of 065° to a buoy Q. It then sails 35 km on a bearing of 140° to a port R.