Vectors - IB Math AI Question Bank

Question 1

HL • Paper 1

Easy

Calculator Permitted

HL • Paper 1

Easy

Calculator Permitted

A rescue drone is at the point $A(2,-1,4)$ and travels directly towards the point $B(8,3,1)$ . Coordinates are measured in metres. The drone travels at a constant speed of $12\ \text{m s}^{-1}$ .

A

Find the vector $\overrightarrow{AB}$ .

[1]

B

Find the unit vector in the direction from $A$ to $B$ .

[2]

C

Find the velocity vector of the drone.

[2]

D

Find the position of the drone after $0.5$ seconds.

[1]

Question 2

HL • Paper 1

Easy

Calculator Permitted

HL • Paper 1

Easy

Calculator Permitted

A line $L$ passes through the points $P(1,-2,3)$ and $Q(5,4,-1)$ .

A

Find a vector equation of $L$ .

[2]

B

Write down parametric equations for $L$ .

[1]

C

Determine whether each of the points $R(7,7,-3)$ and $S(3,1,2)$ lies on $L$ .

[2]

Question 3

HL • Paper 1

Easy

Calculator Permitted

HL • Paper 1

Easy

Calculator Permitted

The positions of two boats, $A$ and $B$ , at time $t$ hours are modelled by

\boldsymbol{r}_A(t)=\begin{pmatrix}2\\5\end{pmatrix}+t\begin{pmatrix}3\\-1\end{pmatrix},\qquad \boldsymbol{r}_B(t)=\begin{pmatrix}14\\8\end{pmatrix}+t\begin{pmatrix}-1\\-2\end{pmatrix}

Distances are measured in kilometres.

A

Find the speed of boat $A$ .

[1]

B

Determine when and where the two boats meet.

[3]

C

Find the distance travelled by boat $A$ before the boats meet.

[2]

Question 4

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

A particle moves in a vertical plane. Its velocity at time $t$ seconds is

\boldsymbol{v}(t)=\begin{pmatrix}4\\6-2t\end{pmatrix}\ \text{m s}^{-1}

At $t=0$ , its position vector is $\begin{pmatrix}1\\-3\end{pmatrix}$ metres.

A

Find the position vector $\boldsymbol{r}(t)$ of the particle.

[3]

B

Determine the time at which the particle reaches its maximum vertical position.

[1]

C

Find the maximum vertical coordinate of the particle.

[1]

D

Find the position of the particle when it reaches its maximum vertical position.

[1]

Question 5

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

A ball is projected from ground level. Its velocity after $t$ seconds is modelled by

\boldsymbol{v}(t)=\begin{pmatrix}8\\12-9.8t\end{pmatrix}\ \text{m s}^{-1}

At $t=0$ , its position is $\begin{pmatrix}0\\0\end{pmatrix}$ . Air resistance is neglected.

A

Find the position vector of the ball at time $t$ .

[2]

B

Determine the positive value of $t$ at which the ball returns to ground level.

[2]

C

Find the horizontal distance travelled before the ball returns to ground level.

[1]

D

Find the speed of the ball immediately before it reaches the ground.

[1]

Question 6

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

A constant force $\boldsymbol{F}=\begin{pmatrix}120\\-40\\30\end{pmatrix}$ newtons moves an object through a displacement $\boldsymbol{d}=\begin{pmatrix}6\\2\\-1\end{pmatrix}$ metres.

A

Calculate the work done by the force.

[2]

B

Find the angle between the force and the displacement. Give your answer in degrees.

[3]

C

Explain what the sign of the work done indicates about the force.

[1]

Question 7

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

The points $A(1,0,2)$ , $B(4,2,1)$ and $C(-1,3,5)$ form a triangle.

A

Find the vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$ .

[2]

B

Find $\overrightarrow{AB}\times\overrightarrow{AC}$ .

[2]

C

Find the area of triangle $ABC$ .

[1]

D

Find a unit vector perpendicular to the plane containing $A$ , $B$ and $C$ .

[1]

Question 8

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

A force $\boldsymbol{F}=\begin{pmatrix}6\\3\\-2\end{pmatrix}$ newtons acts on a straight beam with direction vector $\boldsymbol{b}=\begin{pmatrix}2\\-1\\2\end{pmatrix}$ .

A

Find the scalar component of $\boldsymbol{F}$ acting in the direction of the beam.

[2]

B

Find the magnitude of the component of $\boldsymbol{F}$ perpendicular to the beam.

[2]

C

State whether the component found in part (a) acts with or against the stated direction of $\boldsymbol{b}$ . Give a reason.

[1]

Question 9

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

Two cables exert forces

\boldsymbol{F}_1=\begin{pmatrix}4\\-2\\5\end{pmatrix}\ \text{kN}, \qquad \boldsymbol{F}_2=\begin{pmatrix}-1\\6\\2\end{pmatrix}\ \text{kN}

on a joint.

A

Find the resultant of the two forces.

[1]

B

Find the magnitude of the resultant.

[2]

C

A third cable keeps the joint in equilibrium. Find the force exerted by this cable.

[1]

D

Find the angle between $\boldsymbol{F}_1$ and $\boldsymbol{F}_2$ . Give your answer in degrees.

[2]

Question 10

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

Two autonomous vehicles move for $0\le t\le5$ , where $t$ is measured in minutes. Their position vectors, in kilometres, are

\boldsymbol{r}_A(t)=\begin{pmatrix}0\\4\end{pmatrix}+t\begin{pmatrix}2\\1\end{pmatrix},\qquad \boldsymbol{r}_B(t)=\begin{pmatrix}10\\0\end{pmatrix}+t\begin{pmatrix}-1\\2\end{pmatrix}

A

Find the relative position vector of $B$ from $A$ at time $t$ .

[2]

B

Write down an expression for the square of the distance between the vehicles.

[1]

C

Determine the time at which the distance between the vehicles is a minimum.

[2]

D

Find the minimum distance between the vehicles.

[1]

Question 11

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

Two lines have vector equations

L_1:\ \boldsymbol{r}=\begin{pmatrix}1\\2\\-1\end{pmatrix}+\lambda\begin{pmatrix}2\\-1\\2\end{pmatrix}, \qquad L_2:\ \boldsymbol{r}=\begin{pmatrix}4\\-2\\3\end{pmatrix}+\mu\begin{pmatrix}1\\2\\-2\end{pmatrix}

A

Find the acute angle between $L_1$ and $L_2$ . Give your answer in degrees.

[3]

B

Determine whether the two lines intersect.

[3]

Question 12

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

Two vehicles start moving at the same time. Their positions after $t$ hours are

\boldsymbol{r}_A(t)=\begin{pmatrix}0\\0\end{pmatrix}+t\begin{pmatrix}2\\1\end{pmatrix}, \qquad \boldsymbol{r}_B(t)=\begin{pmatrix}6\\-2\end{pmatrix}+t\begin{pmatrix}-1\\1\end{pmatrix}

Distances are measured in kilometres.

A

Using independent parameters $\lambda$ and $\mu$ , determine the point at which the geometric paths of the vehicles intersect.

[3]

B

Find the time at which each vehicle reaches the intersection point.

[2]

C

State whether the vehicles collide at the intersection point. Give a reason.

[1]

Question 13

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

The position vector of a particle moving in a horizontal plane is

\boldsymbol{r}(t)=\begin{pmatrix}5\cos(0.4t)\\5\sin(0.4t)\end{pmatrix}\ \text{m}

where $t$ is measured in seconds.

A

Find the velocity vector of the particle.

[2]

B

Show that the speed of the particle is constant.

[1]

C

Find the acceleration vector of the particle.

[2]

D

Show that the position vector is perpendicular to the velocity vector for all $t$ .

[1]

Question 14

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

A force $\boldsymbol{F}=\begin{pmatrix}0\\40\\-30\end{pmatrix}$ newtons is applied at a point whose position relative to a pivot is $\boldsymbol{r}=\begin{pmatrix}0.25\\0.10\\0\end{pmatrix}$ metres. The torque vector is defined by $\boldsymbol{\tau}=\boldsymbol{r}\times\boldsymbol{F}$ .

A

Calculate the torque vector.

[2]

B

Find the magnitude of the torque.

[2]

C

Find the angle between $\boldsymbol{r}$ and $\boldsymbol{F}$ . Give your answer in degrees.

[2]

Question 15

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

A straight pipeline is modelled by the line

L:\ \boldsymbol{r}=\begin{pmatrix}1\\-1\\2\end{pmatrix}+\lambda\begin{pmatrix}2\\1\\-2\end{pmatrix}

A monitoring station is located at $P(7,3,1)$ .

A

Let $Q$ be the point on $L$ closest to $P$ . Form an equation in $\lambda$ using the fact that $\overrightarrow{PQ}$ is perpendicular to $L$ .

[2]

B

Hence find the coordinates of $Q$ .

[2]

C

Find the shortest distance from the monitoring station to the pipeline.

[2]

Question 16

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

A particle $A$ moves according to

\boldsymbol{r}_A(t)=\begin{pmatrix}3\cos t\\3\sin t\end{pmatrix}

where $t$ is measured in seconds. A second particle $B$ follows the same motion exactly $2$ seconds later.

A

Write down the position vector $\boldsymbol{r}_B(t)$ .

[1]

B

Find the velocity vector of $B$ at $t=2$ .

[2]

C

Show that the distance between $A$ and $B$ is constant, and find this distance.

[3]

Question 17

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

An aircraft flies in a horizontal plane. East and north are represented by the positive $x$ - and $y$ -directions respectively. The aircraft has an airspeed of $240\ \text{km h}^{-1}$ in the direction $3\boldsymbol{i}+4\boldsymbol{j}$ . A constant wind has velocity $-30\boldsymbol{i}+20\boldsymbol{j}\ \text{km h}^{-1}$ .

A

I.

Find the velocity vector of the aircraft relative to the ground.

[3]

II.

Calculate the ground speed of the aircraft.

[2]

B

I.

Find the displacement of the aircraft after $1.75$ hours.

[2]

II.

The aircraft then returns directly to its starting point with a ground speed of $260\ \text{km h}^{-1}$ . Find its return velocity vector and the return time.

[3]

Question 18

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

A straight railway tunnel passes through the points $A(-2,1,4)$ and $B(4,5,-2)$ . A maintenance shaft is modelled by the line

M:\ \boldsymbol{r}=\begin{pmatrix}7\\7\\-5\end{pmatrix}+\mu\begin{pmatrix}1\\-2\\2\end{pmatrix}

A perspective diagram showing the railway tunnel through labelled points A, B and C, with A, then B, then C in that order along the tunnel, and the maintenance shaft M meeting the tunnel at C, without displaying coordinates or calculated angles.

A

I.

Find a vector equation of the railway tunnel.

[2]

II.

Show that the point $C(7,7,-5)$ lies on the railway tunnel.

[3]

B

Hence find the acute angle at which the maintenance shaft meets the railway tunnel.

[4]

Question 19

HL • Paper 2

Medium

Calculator Permitted

HL • Paper 2

Medium

Calculator Permitted

A towing cable exerts a force of magnitude $500\ \text{N}$ in the direction $4\boldsymbol{i}+\boldsymbol{j}$ . During one stage of the tow, the load has displacement $24\boldsymbol{i}+10\boldsymbol{j}$ metres.

A

I.

Find the force vector.

[2]

II.

Calculate the work done by the cable.

[2]

B

I.

Find the scalar component of the force parallel to the displacement.

[2]

II.

Find the magnitude of the component of the force perpendicular to the displacement and interpret its size.

[3]

Question 20

HL • Paper 3

Medium

Calculator Permitted

HL • Paper 3

Medium

Calculator Permitted

Three environmental sensors are placed at $A(2,-1,3)$ , $B(8,2,0)$ and $C(-1,5,6)$ . A communications hub is to be positioned using the medians of triangle $ABC$ .

Point	x	y	z
A	2	-1	3
B	8	2	0
C	-1	5	6

A

I.

Find $\overrightarrow{AB}$ and $\overrightarrow{AC}$ .

[2]

II.

Find the midpoint $M$ of $BC$ and a vector equation of the median $AM$ .

[2]

B

The hub $G$ is placed two-thirds of the way from $A$ to $M$ . Find $G$ and show that its position vector is $\frac13(\boldsymbol{a}+\boldsymbol{b}+\boldsymbol{c})$ . Hence explain why the same point lies two-thirds of the way along each median.

[4]

Question 21

HL • Paper 3

Medium

Calculator Permitted

HL • Paper 3

Medium

Calculator Permitted

A robotic tool applies the constant force $\boldsymbol{F}=(3,-2,1)$ newtons while moving from $O=(0,0,0)$ to $B=(4,1,3)$ metres. It may move directly or via $A=(1,3,-1)$ .

A schematic, not-to-scale three-dimensional illustration showing points O, A and B, a direct route from O to B, a two-segment route through A, and a constant-force arrow. Do not include coordinate axes or use the drawing to imply the directional meaning of the stated coordinates or force components.

A

I.

Find the work done along the direct route from $O$ to $B$ .

[2]

II.

Find the total work done along the route $OAB$ .

[2]

B

For a constant force $\boldsymbol{F}$ and any sequence of points $P_0,P_1,\ldots,P_n$ , show that the total work from $P_0$ to $P_n$ is independent of the intermediate points. State why this conclusion need not hold if the force changes with position.

[4]

Question 22

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Two survey vessels move for $0\leq t\leq6$ , where $t$ is measured in hours and positions are measured in kilometres:

\boldsymbol{r}_A(t)=\begin{pmatrix}0\\12\end{pmatrix}+t\begin{pmatrix}5\\1\end{pmatrix},\qquad \boldsymbol{r}_B(t)=\begin{pmatrix}18\\0\end{pmatrix}+t\begin{pmatrix}-1\\4\end{pmatrix}

A

AI.

Find the relative position vector of $B$ from $A$ at time $t$ .

[2]

AII.

Show that the square of their separation is $45t^2-288t+468$ .

[2]

B

BI.

Determine the time at which the vessels are closest and find their minimum separation.

[4]

BII.

Safety regulations require the vessels to remain at least $3\ \text{km}$ apart. State whether the regulation is satisfied, with a reason.

[1]

Question 23

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A water droplet moves in a vertical plane. Its velocity at time $t$ seconds is

\boldsymbol{v}(t)=\begin{pmatrix}6\\8-1.6t\end{pmatrix}\ \text{m s}^{-1}

At $t=0$ , its position is $(-3,2)$ metres.

A

I.

Find the position vector $\boldsymbol{r}(t)$ .

[3]

II.

Determine the maximum vertical coordinate reached by the droplet.

[3]

B

The droplet passes through the horizontal level $y=10$ twice.

I.

Find the two horizontal coordinates at which the droplet passes through $y=10$ .

[3]

II.

Find the speed of the droplet when it passes through $y=10$ for the second time. If you did not obtain a second time, use $t=8.873$ .

[2]

Question 24

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A rectangular solar panel has adjacent vertices $A(1,2,0)$ , $B(7,3,-2)$ and $D(2,6,5)$ . Coordinates are measured in metres. The direction from the panel towards the Sun is represented by $\boldsymbol{s}=\begin{pmatrix}2\\-1\\-2\end{pmatrix}$ .

A three-dimensional diagram of a rectangular solar panel with adjacent vertices A, B and D labelled, together with an arrow showing the direction from the panel towards the Sun.

A

I.

Find $\overrightarrow{AB}\times\overrightarrow{AD}$ .

[3]

II.

Hence find the area of the panel and a unit normal vector to it.

[3]

B

I.

Find the acute angle between the Sun's direction and a normal to the panel.

[2]

II.

The effective collecting area is the area of the panel multiplied by the cosine of this angle. Find the effective collecting area.

[2]

Question 25

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A mechanic applies a force of magnitude $180\ \text{N}$ in the direction $3\boldsymbol{j}-4\boldsymbol{k}$ . The position vector from the centre of a bolt to the point of application is

\boldsymbol{r}=0.25\boldsymbol{i}+0.08\boldsymbol{j}\ \text{m}

The torque vector is $\boldsymbol{\tau}=\boldsymbol{r}\times\boldsymbol{F}$ .

A diagram of a wrench turning a bolt, showing the lever-arm vector from the bolt to the point where the force is applied and the three-dimensional force direction.

A

I.

Find the force vector.

[2]

II.

Calculate the torque vector and its magnitude.

[4]

B

Find a unit vector along the axis of rotation indicated by the torque and explain the effect of reversing the force.

[3]

Question 26

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

In a computer-graphics model, a surface has outward normal vector $\boldsymbol{n}=2\boldsymbol{i}-\boldsymbol{j}+2\boldsymbol{k}$ . The direction from the surface towards a light source is $\boldsymbol{s}=\boldsymbol{i}-2\boldsymbol{j}+2\boldsymbol{k}$ . The illumination factor is modelled by the cosine of the acute angle between the two vectors.

A

I.

Find unit vectors in the directions of $\boldsymbol{n}$ and $\boldsymbol{s}$ .

[2]

II.

Calculate the illumination factor and the angle between the normal and the light direction.

[3]

B

A second surface has normal vector $\boldsymbol{m}=k\boldsymbol{i}+\boldsymbol{j}+2\boldsymbol{k}$ . Determine the value of $k$ for which this surface receives no direct illumination, and justify your answer.

[3]

Question 27

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A river is $600\ \text{m}$ wide and flows due east at $1.8\ \text{m s}^{-1}$ . A boat moves through the water at a constant speed of $4.5\ \text{m s}^{-1}$ . The boat starts on the south bank directly opposite a landing point on the north bank. The diagram is schematic and not to scale; the boat's heading is specified in each part.

A schematic, not-to-scale plan view of a river with parallel north and south banks, with the starting point S and landing point L vertically aligned on opposite banks. Show an eastward current arrow and no boat-heading arrow, since the boat's heading is specified in the question parts.

A

I.

Find the boat's velocity relative to the water if it is to travel directly to the landing point.

[3]

II.

Find the time taken to reach the landing point.

[2]

B

The boat instead points due north while maintaining the same speed through the water.

I.

If the boat is pointed due north relative to the water, find the crossing time and the distance east of the landing point at which the boat reaches the north bank.

[3]

II.

Compare the direct-to-landing strategy in part (a) with the due-north strategy in part (b)(i) in terms of crossing time and arrival position.

[2]

Question 28

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A particle $A$ moves around a circle according to

\boldsymbol{r}_A(t)=\begin{pmatrix}8\cos(0.3t)\\8\sin(0.3t)\end{pmatrix}\ \text{m}

A particle $B$ follows the same motion exactly $4$ seconds later.

Circular path with positions A and B at t=4 s.

A

I.

Write down the position vector of $B$ at time $t$ .

[1]

II.

Find the velocity vector and speed of $B$ at $t=4$ .

[3]

B

I.

Show that the distance between $A$ and $B$ is constant, and find this distance.

[3]

II.

Find the first time $t\geq4$ at which $B$ is at the point $(0,8)$ .

[2]

Question 29

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Two inspection pods move through a three-dimensional storage facility. Their positions after $t$ seconds are

\boldsymbol{r}_A(t)=\begin{pmatrix}1\\0\\3\end{pmatrix}+t\begin{pmatrix}2\\1\\-1\end{pmatrix},\qquad \boldsymbol{r}_B(t)=\begin{pmatrix}9\\-5\\0\end{pmatrix}+t\begin{pmatrix}-1\\2\\1\end{pmatrix}

Distances are measured in metres.

A

I.

Find the angle between the velocity vectors of the pods.

[3]

II.

Find the acute angle between their geometric paths.

[1]

B

I.

Show that the relative position of $B$ from $A$ is $\begin{pmatrix}8-3t\\-5+t\\-3+2t\end{pmatrix}$ .

[2]

II.

Determine when the pods are closest and find their positions and separation at that time.

[4]

Question 30

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Two cables exert forces on a joint. Cable 1 exerts a force of magnitude $8\ \text{kN}$ in the direction $2\boldsymbol{i}-\boldsymbol{j}+2\boldsymbol{k}$ . Cable 2 exerts a force of magnitude $10\ \text{kN}$ in the direction $-3\boldsymbol{i}+4\boldsymbol{j}$ . A mast through the joint has direction $\boldsymbol{j}+\boldsymbol{k}$ .

A

I.

Find the two force vectors.

[3]

II.

Find the resultant force and its magnitude.

[3]

B

I.

Find the scalar component of the resultant acting in the direction of the mast.

[2]

II.

Find the magnitude of the component perpendicular to the mast.

[2]

Question 31

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A stationary camera drone is at $D(4,-3,12)$ metres. Its camera is initially directed towards a ground marker $M(10,5,0)$ . The ground is represented by the plane $z=0$ .

A three-dimensional diagram showing the elevated drone D, the ground marker M, the line of sight, and the horizontal ground plane.

A

I.

Find a vector equation of the initial line of sight.

[2]

II.

A transparent vertical screen is represented by $x=7$ . Find where the line of sight passes through the screen and its distance from the drone.

[3]

B

A ground vehicle has position

\boldsymbol{r}_V(t)=\begin{pmatrix}0\\-5\\0\end{pmatrix}+t\begin{pmatrix}2\\2\\0\end{pmatrix},\qquad t\geq0

where $t$ is measured in seconds.

I.

Find the time when the vehicle is closest to the drone.

[3]

II.

Find the minimum distance from the drone to the vehicle. If you did not obtain a time, use $t=1.50$ .

[2]

Question 32

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

An underwater vehicle travels from $A(1,2,-1)$ to $B(13,8,7)$ . Coordinates are measured in metres. The journey takes $6$ minutes. The water current has velocity

\boldsymbol{c}=\begin{pmatrix}1\\-1\\0.5\end{pmatrix}

in metres per minute. The vehicle maintains a constant velocity relative to the water.

A three-dimensional diagram showing the underwater vehicle travelling from A to B, together with separate arrows for its velocity relative to the water, the current velocity, and the resultant ground velocity.

A

I.

Find the displacement vector $\overrightarrow{AB}$ and the velocity of the vehicle relative to the ground.

[2]

II.

Hence find the velocity and speed of the vehicle relative to the water.

[2]

B

Suppose instead that the current is $k\boldsymbol{c}$ , where $k\geq0$ , while the journey time and route remain unchanged. Determine the value of $k$ for which the magnitude of the required velocity relative to the water is minimized, and find this minimum speed.

[4]

Question 33

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Two straight mining tunnels are modelled by

L_1:\ \boldsymbol{r}=\begin{pmatrix}1\\2\\0\end{pmatrix}+\lambda\begin{pmatrix}2\\-1\\1\end{pmatrix},\qquad L_2:\ \boldsymbol{r}=\begin{pmatrix}7\\-1\\3\end{pmatrix}+\mu\begin{pmatrix}-1\\2\\0\end{pmatrix}

Tunnel	Point	Direction	Vector equation
L1	(1, 2, 0)	(2, -1, 1)	r = (1, 2, 0) + λ(2, -1, 1)
L2	(7, -1, 3)	(-1, 2, 0)	r = (7, -1, 3) + μ(-1, 2, 0)

A

I.

Determine the point at which the tunnels intersect.

[2]

II.

Find the acute angle between the tunnels. Give your answer in degrees.

[2]

B

A third tunnel through the intersection is to bisect the acute angle between $L_1$ and $L_2$ . Determine a vector equation for this tunnel and justify that its direction bisects the angle.

[4]

Question 34

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Two aircraft fly at constant altitude. Their positions, in kilometres, are

\boldsymbol{r}_A(t)=\begin{pmatrix}0\\0\end{pmatrix}+t\begin{pmatrix}4\\1\end{pmatrix},\qquad \boldsymbol{r}_B(t)=\begin{pmatrix}10\\-2\end{pmatrix}+t\begin{pmatrix}-1\\3\end{pmatrix}

where $t$ is measured in minutes and $0\leq t\leq4$ . A separation below $2$ km requires an alert.

Plan-view paths of aircraft A and B with initial positions marked and a 2 km safety circle around A(0).

A

I.

Find an expression for the relative position of $B$ from $A$ and hence for the square of their separation.

[2]

II.

Determine when the aircraft are closest and whether an alert is required.

[2]

B

For two objects with relative position $\boldsymbol{p}+t\boldsymbol{v}$ , derive a formula for the unrestricted time of closest approach. State the additional checks needed when the model applies only for $0\leq t\leq T$ , and explain how an actual collision can be distinguished from a close approach.

[4]

Question 35

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Two inspection robots follow straight routes across a factory floor. Robot $A$ starts at time $t=0$ and has position

\boldsymbol{r}_A(t)=t\begin{pmatrix}2\\1\end{pmatrix}

Robot $B$ begins at $(8,0)$ and moves with velocity $\begin{pmatrix}-1\\2\end{pmatrix}$ once activated.

Plan-view plot of the two robot routes and their start points; the routes cross geometrically without marking the intersection coordinates.

A

I.

Using independent route parameters, find the intersection point of the two geometric paths.

[2]

II.

If both robots are activated at $t=0$ , determine whether they collide at the intersection.

[2]

B

Robot $B$ is instead activated $h$ minutes after robot $A$ . Write a time-shifted position model for $B$ and determine the value of $h$ that causes a collision. Generalize your result in terms of the two travel times to an intersection.

[4]

Question 36

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A camera moves around an elliptical track. Its position after $t$ seconds is

\boldsymbol{r}(t)=\begin{pmatrix}4\cos(0.5t)\\3\sin(0.5t)\end{pmatrix}

in metres. For part (b), take the first circuit to mean $0\leq t<4\pi$ , with $t$ measured in seconds.

Ellipse track with a moving point and tangent line.

A

I.

Find the velocity vector and the speed of the camera.

[2]

II.

Find the minimum and maximum speeds during one complete circuit.

[2]

B

During the first complete circuit, determine all times when the position vector is perpendicular to the velocity vector. Interpret these times geometrically and explain why the speed is not constant even though the path is closed and periodic.

[4]

Question 37

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A parallelogram-shaped solar panel has adjacent edge vectors

\boldsymbol{u}=\begin{pmatrix}4\\0\\1\end{pmatrix},\qquad \boldsymbol{v}=\begin{pmatrix}1\\3\\1\end{pmatrix}

in metres. Sunlight travels parallel to $\boldsymbol{s}=(2,-1,2)$ .

A three-dimensional diagram of a parallelogram solar panel with adjacent edge vectors labeled $\boldsymbol{u}$ and $\boldsymbol{v}$, a normal vector, and parallel sunlight rays in direction $\boldsymbol{s}$. Do not show numerical components for any vector; use the stem for all vector components.

A

I.

Find a vector perpendicular to the panel and hence find its area.

[2]

II.

Find a unit normal to the panel with a positive vertical component.

[2]

B

The effective collecting area is the area of the panel projected perpendicular to the sunlight. Find this projected area and determine the greatest projected area attainable by rotating the panel without changing its shape.

[4]

Question 38

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

At an airport, the wind velocity is $\boldsymbol{w}=(8,-6,2)\ \text{m s}^{-1}$ . A sloping runway has direction vector $\boldsymbol{d}=(2,1,2)$ in the stated take-off direction.

A coordinate-consistent three-dimensional airport diagram showing the runway direction $\boldsymbol{d}=(2,1,2)$, the wind vector $\boldsymbol{w}=(8,-6,2)$ oriented according to the displayed coordinate axes, and its decomposition into $\boldsymbol{w}_{\parallel}$ and $\boldsymbol{w}_{\perp}$. The vector $\boldsymbol{w}_{\parallel}$ is collinear with $\boldsymbol{d}$; $\boldsymbol{w}_{\perp}$ joins the endpoint of $\boldsymbol{w}_{\parallel}$ to the endpoint of $\boldsymbol{w}$ and is perpendicular to the runway. If an inset for part (b) is shown, label the angle $\theta$ consistently with the stem, indicate that $\boldsymbol{q}=(\cos\theta,\sin\theta,0)$ may point in any horizontal direction, and show the maximizing direction $\boldsymbol{q}=(0.8,-0.6,0)$ in the appropriate quadrant.

A

I.

Find the signed scalar component of the wind in the runway direction and interpret its sign.

[2]

II.

Find the magnitude of the component perpendicular to the runway.

[2]

B

A new runway must be horizontal and may have any unit direction $\boldsymbol{q}=(\cos\theta,\sin\theta,0)$ . Determine the direction that maximizes the tailwind component. Find this maximum component and the remaining perpendicular component of the wind.

[4]

Question 39

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A triangular fabric panel has vertices $A=(0,0,0)$ , $B=(4,0,0)$ and $C=(1,k,2)$ , where $k$ is a real design parameter. Coordinates are measured in metres.

Point	x [m]	y [m]	z [m]
A	0	0	0
B	4	0	0
C	1	k	2

A

I.

Find $\overrightarrow{AB}\times\overrightarrow{AC}$ in terms of $k$ .

[2]

II.

Hence obtain an expression for the area of the panel.

[2]

B

The required panel area is $10\ \text{m}^2$ . Determine the possible values of $k$ and the angle $BAC$ for either design. Then determine the minimum possible area over all real values of $k$ and explain the geometry of the minimizing design.

[4]

Question 40

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Robot $A$ starts at time $t=0$ minutes and has position

\boldsymbol{r}_A(t)=\begin{pmatrix}2\\-4\end{pmatrix}+t\begin{pmatrix}3\\2\end{pmatrix}

Robot $B$ starts at time $t=2$ minutes from $(20,0)$ and then moves with velocity $\begin{pmatrix}-2\\1\end{pmatrix}$ . Positions are measured in kilometres.

A

I.

Using independent parameters, find the point where the geometric paths of the robots intersect.

[4]

II.

Determine whether the robots collide at this point.

[2]

B

For $t\geq2$ , determine when the robots are closest and find their minimum separation.

[5]

Question 41

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A triangular sail has vertices $A(1,0,2)$ , $B(5,2,1)$ and $C(2,k,6)$ , where $k$ is a real number. Coordinates are measured in metres.

A

I.

Find $\overrightarrow{AB}$ and $\overrightarrow{AC}$ in terms of $k$ .

[2]

II.

Show that $\overrightarrow{AB}\times\overrightarrow{AC}=\begin{pmatrix}k+8\\-17\\4k-2\end{pmatrix}$ .

[3]

B

The area of the sail is $\dfrac{\sqrt{425}}{2}\ \text{m}^2$ . Therefore, $\left\lvert\overrightarrow{AB}\times\overrightarrow{AC}\right\rvert^2=425$ .

I.

Find the possible values of $k$ .

[3]

II.

The chosen orientation requires the $\boldsymbol{k}$ -component of $\overrightarrow{AB}\times\overrightarrow{AC}$ to be positive. Determine $k$ and justify your choice.

[2]

Question 42

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Two pipes have centre-lines

L_1:\ \boldsymbol{r}=\lambda\begin{pmatrix}1\\1\\0\end{pmatrix},\qquad L_2:\ \boldsymbol{r}=\begin{pmatrix}1\\0\\2\end{pmatrix}+\mu\begin{pmatrix}0\\1\\1\end{pmatrix}

A shortest connecting pipe is to be installed between them.

Qualitative $u$-$v$ schematic only (not to scale); its plotted coordinates are not a faithful projection of the three-dimensional lines or connector. The exact endpoints, $P=(0,0,0)$ and $Q=(1,-1,1)$, are determined from the vector equations.

A

I.

Let $P$ be a point on $L_1$ and $Q$ a point on $L_2$ . Write $\overrightarrow{PQ}$ in terms of $\lambda$ and $\mu$ .

[2]

II.

Use perpendicularity to determine the endpoints of the shortest connecting pipe.

[2]

B

Find the length of the connecting pipe. Then verify the result using the cross product of the direction vectors and the displacement between any one point on each line.

[4]

Question 43

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A package is launched from ground level with initial velocity $\begin{pmatrix}c\\10\end{pmatrix}\ \text{m s}^{-1}$ , where $c>0$ . Its acceleration is $\begin{pmatrix}0\\-10\end{pmatrix}\ \text{m s}^{-2}$ . A vertical barrier is $9$ m from the launch point and has height $3$ m.

A schematic side-view diagram showing the launch point, horizontal and vertical coordinate directions, and a vertical barrier 9 metres from the launch point with height 3 metres. No trajectory is shown, since the trajectory depends on the variable $c$.

A

I.

Show that the position of the package is

\boldsymbol{r}(t)=\begin{pmatrix}ct\\10t-5t^2\end{pmatrix}

[2]

II.

For $c=6$ , determine whether the package clears the barrier.

[2]

B

Determine the complete interval of values of $c$ for which the package reaches the barrier and passes strictly above its 3 m height before returning to ground level.

[4]

Question 44

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

In a computer graphics model, a surface has normal vector $\boldsymbol{n}=(2,-1,2)$ . A light source lies in the direction $\boldsymbol{s}=(1,-2,2)$ from the surface. The diffuse illumination is modelled by

I=I_0\max\left(0,\widehat{\boldsymbol{n}}\cdot\widehat{\boldsymbol{s}}\right)

A computer-graphics style diagram showing a flat surface, its normal vector n, the direction s toward a light source, and the angle between the two vectors.

A

I.

Find the unit vectors $\widehat{\boldsymbol{n}}$ and $\widehat{\boldsymbol{s}}$ .

[2]

II.

Find $I$ as a fraction of $I_0$ .

[2]

B

The light source is moved so that its direction is $\boldsymbol{s}(k)=(1,-2,k)$ , where $k$ is real. Determine the value of $k$ that maximizes the illumination and find the maximum value of $I/I_0$ .

[4]

Question 45

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A force $\boldsymbol{F}=(0,40,-30)$ newtons is applied to a handle. The point of application is initially at position $\boldsymbol{r}=(0.3,0,0)$ metres relative to the pivot. The torque is $\boldsymbol{\tau}=\boldsymbol{r}\times\boldsymbol{F}$ .

A three-dimensional pivot-and-handle diagram showing the position vector $\boldsymbol r$ from the pivot and the applied force $\boldsymbol F$. Include one clearly placed label for $\boldsymbol r=(0.3,0,0)\ \text{m}$ with a leader pointing to the handle. Omit the torque-axis annotation and circular arrow.

A

I.

Find the torque vector.

[2]

II.

Find the torque magnitude and explain why it is as large as possible for a handle of length $0.3$ m and a force of magnitude $50$ N.

[2]

B

The handle is rotated in the $xy$ -plane so that $\boldsymbol r(\phi)=0.3(\cos\phi,\sin\phi,0)$ . Find the maximum and minimum possible torque magnitudes and the corresponding orientations of the handle.

[4]

Question 46

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Two satellites are modelled over a short time interval by

\boldsymbol{r}_A(t)=t\begin{pmatrix}1\\2\\0\end{pmatrix},\qquad \boldsymbol{r}_B(t)=\begin{pmatrix}6\\-3\\4\end{pmatrix}+t\begin{pmatrix}-1\\1\\-1\end{pmatrix}

where distance is measured in kilometres and time in minutes. A warning is active whenever their separation is at most $6$ km.

Separation-squared curve with warning limit.

A

I.

Find the relative position of satellite $B$ from satellite $A$ and an expression for the square of their separation.

[2]

II.

Find the time and distance of closest approach.

[2]

B

Determine the interval of time for which the warning is active and find its duration. Explain why the midpoint of this interval is the time of closest approach.

[4]

Question 47

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A cable supporting an aerial platform has direction vector $\boldsymbol{d}=(3,4,12)$ . The platform's weight is represented by the force $\boldsymbol{W}=(0,0,-120)$ newtons.

A three-dimensional support-cable diagram showing the cable direction, the downward weight vector, and the components of the weight. Label the component parallel to the cable as "component parallel to cable" and ensure its leader line terminates on the parallel component vector. Show exactly one clearly labelled x-axis, y-axis and z-axis at the origin.

A

I.

Find the signed scalar component of the weight in the stated cable direction.

[2]

II.

Find the magnitude of the component of the weight perpendicular to the cable.

[2]

B

A redesigned cable has direction vector $(3,4,h)$ , where $h>0$ . Safety regulations require the component of the weight perpendicular to the cable to be at most $30$ N. Determine the least possible value of $h$ and interpret this design requirement geometrically.

[4]

Question 48

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Two particles move in a horizontal plane for $0\leq t\leq4$ , where positions are measured in metres and time in seconds:

\boldsymbol{r}_A(t)=\begin{pmatrix}t^2\\4t\end{pmatrix},\qquad \boldsymbol{r}_B(t)=\begin{pmatrix}12-2t\\t^2\end{pmatrix}

A

I.

Find the velocity vectors of the two particles.

[2]

II.

Show that the square of the distance between the particles is

D^2(t)=(12-2t-t^2)^2+(t^2-4t)^2

[3]

B

I.

Determine the time at which the particles are closest. You should compare any stationary point with the endpoints of the interval.

[4]

II.

Find the minimum distance between the particles and verify that the relative position is perpendicular to the relative velocity at this time. If you did not obtain a time, use the unrounded value $t=2.69623\ \text{s}$ .

[2]