3.10 Vectors questions for IB Maths AI

48 questions on Vectors, written by Augment in the style of IB exam papers. Six of them are below in full, to answer on paper.

IB Mathematics: Applications and Interpretation
Part of 3 Geometry and trigonometry
HL only

Questions by paper and level

Paper	Questions
Paper 1	25
Paper 2	21
Paper 3	2
Every paper	48

Counted on 10 November 2026

Questions to try

Written by Augment in the style of IB exams, not past papers. Answer them on paper, then check each one against its markscheme with a free account.

Question 1 · AI-00043Paper 1 · 6 marks

A ferry leaves a pier at 09:00. Its position vector, relative to a lighthouse, $t$ hours after 09:00 is given by

r = (\begin{array}{c} 2 \\ 5 \end{array}) + t (\begin{array}{c} 6 \\ - 8 \end{array})

where distances are measured in kilometres. The unit vector $(\begin{array}{c} 1 \\ 0 \end{array})$ points east and $(\begin{array}{c} 0 \\ 1 \end{array})$ points north.

(a)
Find the speed of the ferry.
2 marks
(b)
Find the position vector of the ferry at 10:30.
2 marks
(c)
Find the bearing on which the ferry is travelling.
2 marks

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Question 2 · AI-00281Paper 2 · 15 marks

Oliver is planning a square courtyard and uses an Argand diagram as a plan, with $1$ unit representing $1$ metre. One corner of the courtyard is at the origin $O$ , and a second corner is at the point $A$ , which represents $z = 6 + 2 i$ .

The third corner, $B$ , represents $i z$ and the fourth corner, $C$ , represents $z + i z$ .

(a)
Find $i z$ and $z + i z$ in the form $a + b i$ .
2 marks
(b)
(i) Show that $∣ z ∣ = ∣ i z ∣$ .
(ii) Show that $O A$ is perpendicular to $O B$ .
4 marks
(c)
Find the area of the courtyard.
2 marks
(d)
A straight path runs from $O$ to $C$ . Find the length of the path.
2 marks
(e)
Find the angle between the path $O C$ and the side $O A$ .
2 marks
(f)
A fountain is placed at the centre of the courtyard, the point $F$ .
(i) Find the complex number represented by $F$ .
(ii) Find the distance from the fountain to $A$ .
3 marks

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Question 3 · AI-01630Paper 3 · 28 marks

In this question you will use vectors to decide how a research vessel can get close to a migrating whale.

Marine biologists at a research station in Hermanus, South Africa, are tracking a humpback whale that has been fitted with a satellite tag. Positions are given relative to the harbour, $O$ , with the $x$ -axis pointing east and the $y$ -axis pointing north. Distances are in kilometres and $t$ is the time in hours after 06:00.

The position vector of the whale at time $t$ is modelled by

r_{W} = (\begin{array}{c} 20 \\ 35 \end{array}) + t (\begin{array}{c} 6 \\ - 8 \end{array}) .

(a)(i)
Find the speed of the whale.
1 mark
(a)(ii)
Find the bearing on which the whale is travelling.
2 marks
(b)
Find the shortest distance between the whale and the harbour, and the time at which this occurs.
3 marks
(c)
At 06:00 the research vessel leaves the harbour with velocity $(\begin{array}{c} 12 \\ 9 \end{array})$ km h⁻¹.
Write down the position vector, $r_{V}$ , of the vessel at time $t$ .
1 mark
(d)
Show that the displacement vector from the vessel to the whale at time $t$ is $(\begin{array}{c} 20 - 6 t \\ 35 - 17 t \end{array})$ .
2 marks
(e)
Find the minimum distance between the vessel and the whale, and the time at which it occurs.
4 marks
(f)
The vessel's sonar can detect the whale when the distance between them is 10 km or less.
Find, to the nearest minute, the length of time for which the sonar can detect the whale.
3 marks
(g)(i)
On another day the whale follows the same path at the same times. This time the captain wants to meet the whale. The vessel leaves the harbour at 06:00 and travels in a straight line at a constant speed of 15 km h⁻¹.
The vessel meets the whale at time $t = T$ .
Explain why $∣ r_{W} (T) ∣ = 15 T$ , and hence show that $125 T^{2} + 320 T - 1625 = 0$ .
3 marks
(g)(ii)
Hence find the time at which the vessel meets the whale.
2 marks
(g)(iii)
Find the bearing on which the vessel must travel.
3 marks
(h)
Find the acute angle between the vessel's path in part (g) and the whale's path.
2 marks
(i)
State two assumptions made in modelling the motion of the whale and the vessel.
2 marks

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Question 4 · AI-00044Paper 1 · 6 marks

A survey drone flies in a straight line at constant velocity. Relative to its launch pad, it is at the point A $(4, - 2, 0.5)$ at time $t = 0$ and at the point B $(36, 22, 10.1)$ when $t = 8$ , where $t$ is measured in seconds and distances in metres. The $z$ -axis is vertical.

(a)
Find the velocity vector of the drone.
2 marks
(b)
Find the speed of the drone.
2 marks
(c)
The drone continues on the same path. Find the time at which it reaches a height of 25 metres.
2 marks

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Question 5 · AI-00335Paper 2 · 16 marks

A vertical pole $O T$ supports the centre of a festival marquee. Its base $O$ is at the origin, the $x$ -axis points east, the $y$ -axis points north and the $z$ -axis points vertically upwards. Distances are in metres. The top of the pole is $T (0, 0, 4.8)$ .

Three straight ropes run from $T$ to pegs $A (3.6, 2.0, 0)$ , $B (- 3.6, 2.0, 0)$ and $C (0, - 3.6, 0)$ on the horizontal ground, as shown in the diagram.

A vertical pole OT stands on horizontal ground with O at its base and T at its top. Three straight ropes run from T to pegs A, B and C on the ground. — Diagram not to scale.

(a)
Write down $\vec{T A}$ .
1 mark
(b)
Find the length of the rope $T A$ .
2 marks
(c)
Find a unit vector in the direction of $\vec{T A}$ .
2 marks
(d)
The tension in rope $T A$ is $150$ newtons, so the force $F_{A}$ exerted on $T$ by this rope has magnitude $150 N$ and acts in the direction of $\vec{T A}$ . Find $F_{A}$ .
2 marks
(e)
The tension in rope $T B$ is also $150$ newtons. Write down the force $F_{B}$ exerted on $T$ by rope $T B$ .
1 mark
(f)
Find $F_{A} + F_{B}$ .
2 marks
(g)
The tension in rope $T C$ is $T_{C}$ newtons. The pole stays upright when the total force exerted on $T$ by the three ropes has no horizontal component. Find the value of $T_{C}$ for which this happens.
4 marks
(h)
Hence find the magnitude of the total force exerted on $T$ by the three ropes.
2 marks

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Question 6 · AI-01826Paper 3 · 28 marks

In this question you will use three-dimensional geometry and vectors to design the roof of a house.

Haruto is an architect designing a house in Sapporo, Japan. The house has a rectangular base $A B C D$ , with $A B = 12$ m and $B C = 8$ m. Its roof is a hipped roof: four sloping faces meet at a horizontal ridge $[E F]$ , as shown in the following diagram.

Hipped roof on a horizontal rectangular base ABCD with AB = 12 m and BC = 8 m. A horizontal ridge EF lies above the base; sloping edges AE, DE, BF and CF join the ridge to the corners. Edges AD, DC and DE are hidden (dashed). — Diagram not to scale.

Haruto sets up coordinates in metres with $A$ at the origin, the $x$ -axis along $[A B]$ , the $y$ -axis along $[A D]$ and the $z$ -axis vertically upwards. The base is horizontal, so $B (12, 0, 0)$ , $C (12, 8, 0)$ and $D (0, 8, 0)$ . The ridge is at height $h$ metres, with $E (3, 4, h)$ and $F (9, 4, h)$ .

In his first design, $h = 3$ .

(a)
Write down the vector $\vec{A E}$ .
1 mark
(b)
Find the length of the edge $[A E]$ .
2 marks
(c)
Find the angle between the edge $[A E]$ and the horizontal base.
3 marks
(d)
The roof covering must be laid at a pitch of at least 30°, where the pitch of a face is the angle between that face and the horizontal.
Show that the face $A B F E$ meets this requirement.
3 marks
(e)
Use a vector product to find the area of the triangular face $A D E$ .
4 marks
(f)
Find the total area of the four faces of the roof.
3 marks
(g)
The space under the roof is divided into a triangular prism, between the vertical planes $x = 3$ and $x = 9$ , and two pyramids, one at each end.
Show that the volume of the space under the roof is $40 h$ m³.
3 marks
(h)
Show that the total area of the roof, $R$ m², is given by $R (h) = 18 \sqrt{h^{2} + 16} + 8 \sqrt{h^{2} + 9}$ .
2 marks
(i)
The owners ask for 160 m³ of space under the roof.
Find the percentage increase in the area of the roof compared with the first design.
3 marks
(j)
Because of heavy snowfall in Sapporo, the pitch of the faces $A B F E$ and $D C F E$ must be between 30° and 45° inclusive. Find the range of possible volumes of the space under the roof.
3 marks
(k)
State one assumption Haruto has made in calculating the volume of the space under the roof.
1 mark

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