3.12 Vectors in kinematics questions for IB Maths AI

33 questions on Vectors in kinematics, 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	15
Paper 2	17
Paper 3	1
Every paper	33

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-00338Paper 2 · 17 marks

A delivery company uses drones. A drone takes off vertically from its depot and, at the point $A (0, 0, 60)$ , begins to fly horizontally in a straight line towards the point $H (840, - 630, 60)$ above a customer's house. The $x$ -axis points east, the $y$ -axis points north and the $z$ -axis points vertically upwards. Distances are in metres and time is in seconds.

(a)
Write down $\vec{A H}$ .
1 mark
(b)
Find the distance $A H$ .
2 marks
(c)
Find a unit vector in the direction of $\vec{A H}$ .
2 marks
(d)
On a calm day the drone flies from $A$ to $H$ at a constant speed of $15 {m s}^{- 1}$ .
(i) Find the velocity vector of the drone.
(ii) Find the time taken to fly from $A$ to $H$ .
3 marks
(e)
On another day a wind blows with velocity $(\begin{array}{c} - 1.5 \\ 2.0 \\ 0 \end{array}) {m s}^{- 1}$ . The drone leaves $A$ with the same velocity relative to the air as in part (d)(i) and keeps it for $70$ seconds. Find how far the drone is from $H$ at the end of the $70$ seconds.
4 marks
(f)
To reach $H$ in this wind, the drone must keep its speed of $15 {m s}^{- 1}$ relative to the air but change its direction so that its velocity relative to the ground is $k (\begin{array}{c} 0.8 \\ - 0.6 \\ 0 \end{array})$ , where $k > 0$ .
(i) Find the value of $k$ .
(ii) Hence find the time taken to fly from $A$ to $H$ in this wind.
5 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-00420Paper 2 · 19 marks

An air traffic controller at a radar station, O, monitors a light aircraft, A, and a weather balloon, B. Relative to O, the $x$ -axis points east, the $y$ -axis points north and the $z$ -axis is vertical, with distances in kilometres. The positions of the aircraft and of the balloon $t$ minutes after 15:00 are given by

r_{A} = (\begin{array}{c} - 30 \\ 12 \\ 3 \end{array}) + t (\begin{array}{c} 4 \\ - 1.5 \\ - 0.1 \end{array}) and r_{B} = (\begin{array}{c} 2 \\ - 4 \\ 0.5 \end{array}) + t (\begin{array}{c} 0.2 \\ 0.1 \\ 0.05 \end{array}) .

(a)
Find the speed of the aircraft in km h $^{- 1}$ .
3 marks
(b)
By making two statements, describe the motion of the balloon.
2 marks
(c)(i)
Find the time at which the aircraft and the balloon are at the same height.
2 marks
(c)(ii)
Hence show that the aircraft and the balloon do not collide.
4 marks
(d)
Regulations require aircraft to remain more than 3 km from a weather balloon. Determine whether the aircraft breaks this regulation.
5 marks
(e)
Find the angle between the directions of motion of the aircraft and the balloon.
3 marks

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Question 6 · AI-00045Paper 1 · 6 marks

Two fishing boats, the Marisol and the Kestrel, move with constant velocities. Their position vectors, in kilometres, $t$ hours after 06:00 are

r_{M} = (\begin{array}{c} - 4 \\ 6 \end{array}) + t (\begin{array}{c} 5 \\ 1 \end{array}) and r_{K} = (\begin{array}{c} 10 \\ - 8 \end{array}) + t (\begin{array}{c} - 2 \\ 6 \end{array}) .

(a)
Show that the paths of the two boats cross, and find the coordinates of the point where they cross.
4 marks
(b)
Determine whether the boats collide. Justify your answer.
2 marks

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