3.11 Vector equations of lines questions for IB Maths AI

30 questions on Vector equations of lines, 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	16
Paper 2	13
Paper 3	1
Every paper	30

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-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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Question 2 · AI-00737Paper 2 · 17 marks

A radar station $R$ is at the origin. The $x$ -axis points east, the $y$ -axis points north and the $z$ -axis points vertically upwards, with distances in kilometres. At 14:00 an aircraft is detected at the point $P (30, - 12, 3.1)$ . It is descending in a straight line in the direction of the vector $d = (\begin{array}{c} - 10 \\ 4 \\ - 0.9 \end{array})$ .

(a)
Write down a vector equation of the line along which the aircraft is flying.
2 marks
(b)
Show that the aircraft passes directly above the radar station.
3 marks
(c)
Find the height of the aircraft when it is directly above the radar station.
1 mark
(d)
Find the angle at which the aircraft is descending, measured from the horizontal.
3 marks
(e)
If the aircraft continued on this line, find the coordinates of the point where it would reach ground level, $z = 0$ .
2 marks
(f)
The summit of a hill is at $H (8, - 2, 1.6)$ . Regulations require aircraft to stay at least $2$ km from the summit. Determine whether the aircraft obeys this regulation.
4 marks
(g)
The parameter $λ$ is the time, in minutes, after 14:00. Find the speed of the aircraft in ${km h}^{- 1}$ .
2 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-00542Paper 1 · 6 marks

A chairlift cable runs in a straight line from the base station $B (120, 40, 1650)$ to the top station $T (860, 520, 2210)$ . The $x$ -axis points east, the $y$ -axis points north and $z$ is the altitude, with all distances in metres.

(a)
Find $\vec{B T}$ .
1 mark
(b)
Write down a vector equation of the line along which the cable runs.
2 marks
(c)
A support tower holds the cable at an altitude of $1930$ m. Find the coordinates of the point where the cable meets the top of the tower.
3 marks

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Question 5 · AI-00738Paper 2 · 16 marks

An adventure park has two straight zip lines. Relative to an origin at ground level, with the $z$ -axis vertically upwards and distances in metres, zip line 1 runs from $A (0, 0, 40)$ to $B (180, 60, 10)$ and zip line 2 runs from $C (20, 100, 35)$ to $D (160, - 20, 5)$ .

(a)
Find $\vec{A B}$ and $\vec{C D}$ .
2 marks
(b)
Write down a vector equation for each zip line, using the parameter $s$ for zip line 1 and $t$ for zip line 2.
2 marks
(c)
Find the length of zip line 1.
2 marks
(d)
Seen from above, the two zip lines cross. Find the values of $s$ and $t$ at which the $x$ - and $y$ -coordinates of the two lines are equal, and hence show that the zip lines do not intersect.
5 marks
(e)
Safety rules require a vertical gap of at least $5$ m where one zip line passes over the other. Determine whether the park meets this rule.
2 marks
(f)
Find the acute angle between the directions of the two zip lines.
3 marks

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Question 6 · AI-00543Paper 1 · 6 marks

In a science museum, a laser beam travels along the line

r = (\begin{array}{c} 2 \\ - 1 \\ 3 \end{array}) + λ (\begin{array}{c} 4 \\ 2 \\ - 1 \end{array}),

where distances are in metres. Two light sensors are placed at $P (14, 5, 0)$ and $Q (- 6, - 5, 5.5)$ .

(a)
Write down the equation of the beam in parametric form.
1 mark
(b)
Show that the beam hits the sensor at $P$ .
2 marks
(c)
Determine whether the beam hits the sensor at $Q$ .
3 marks

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