Free Vectors & kinematics questions for IB Mathematics: Applications & Interpretation (SL & HL), HL only, part of Geometry & trigonometry. Each question has the answer and the marking points an examiner would look for.
An aircraft’s position is (x, y, z) km, where x and y are displacements east and north of an airport and z is the height. At 13:00 it is at (40, 20, 6) and moves with constant velocity −200−100−15 km h−1. Let t be the time in hours after 13:00.
(a)
Write down a vector equation for the position r of the aircraft.
[2]
(b)
Verify that the aircraft would pass directly over the airport, and find its height at that time.
[4]
(c)
Find the time when the aircraft is 4 km above the ground, and its distance from the airport then.
[4]
(d)
The aircraft now adjusts its descent so that it lands at (0, 0, 0), keeping the same horizontal velocity −200−100. Find the new vertical component of velocity.
[4]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
r = 40206 + t−200−100−15
position + t × velocity (M1) r = 40206 + t−200−100−15A1
2
(b)
t = 0.200; 3.00 km
x = 0 ⇒ t = 0.200 A1 y = 0 at the same t R1 substituting into z (M1) 3.00 km A1
4
(c)
t = 0.133 h; 15.4 km
6 − 15t = 4 (M1) t = 0.133 A1 position and distance (M1) 15.4 km A1
4
(d)
−60.0 km h−1
time to reach the airport horizontally (M1) 0.0667 h (A1) −4 ÷ time (M1) −60.0 km h−1A1
4
Question 2Structured · 6 marks
A boat leaves point P with position vector −2−3 km and travels with constant velocity 53 km h−1. A lighthouse is at L(16, 2.8).
(a)
Find the speed of the boat.
[2]
(b)
Find the time at which the boat is due north of the lighthouse, and the distance between them at that time.
[4]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
5.83 km h−1
|53| (M1) 5.83 km h−1A1
2
(b)
t = 3.60 h; 5.00 km
−2 + 5t = 16 (M1) t = 3.60 A1 position (16, 7.80) (A1) 5.00 km A1
4
Question 3Structured · 4 marks
A boat’s position vector t hours after noon is r = −14 + t−1−1 km.
(a)
Find the speed of the boat.
[2]
(b)
Find the position of the boat at 2 pm.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
1.41 km h−1
magnitude of velocity (M1) 1.41 A1
2
(b)
(−3, 2)
substitute t = 2 (M1) (−3, 2) A1
2
Question 4Structured · 14 marks
An aircraft’s position is (x, y, z) km, where x and y are displacements east and north of an airport and z is the height. At 13:00 it is at (30, 10, 5) and moves with constant velocity −150−50−15 km h−1. Let t be the time in hours after 13:00.
(a)
Write down a vector equation for the position r of the aircraft.
[2]
(b)
Verify that the aircraft would pass directly over the airport, and find its height at that time.
[4]
(c)
Find the time when the aircraft is 4 km above the ground, and its distance from the airport then.
[4]
(d)
The aircraft now adjusts its descent so that it lands at (0, 0, 0), keeping the same horizontal velocity −150−50. Find the new vertical component of velocity.
[4]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
r = 30105 + t−150−50−15
position + t × velocity (M1) r = 30105 + t−150−50−15A1
2
(b)
t = 0.200; 2.00 km
x = 0 ⇒ t = 0.200 A1 y = 0 at the same t R1 substituting into z (M1) 2.00 km A1
4
(c)
t = 0.0667 h; 21.5 km
5 − 15t = 4 (M1) t = 0.0667 A1 position and distance (M1) 21.5 km A1
4
(d)
−30.0 km h−1
time to reach the airport horizontally (M1) 0.133 h (A1) −4 ÷ time (M1) −30.0 km h−1A1
4
Question 5Structured · 6 marks
A boat leaves point P with position vector −75 km and travels with constant velocity 63 km h−1. A lighthouse is at L(12, 18.5).
(a)
Find the speed of the boat.
[2]
(b)
Find the time at which the boat is due south of the lighthouse, and the distance between them at that time.
[4]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
6.71 km h−1
|63| (M1) 6.71 km h−1A1
2
(b)
t = 3.17 h; 4.00 km
−7 + 6t = 12 (M1) t = 3.17 A1 position (12, 14.5) (A1) 4.00 km A1
4
Question 6Structured · 4 marks
A boat’s position vector t hours after noon is r = 01 + t−22 km.
Riyaz Shaikh has mentored IGCSE, IB and A Level students for 15+ years, one-on-one, online and in Mumbai. The hardest questions on this site are the ones he uses to find a student's weak points.
Original IB-style practice questions written for mentorriyaz.in. These are not IB past paper questions, and this site is not affiliated with or endorsed by the International Baccalaureate Organization. Updated 2026-11-04. Paper generator · mentorriyaz.in