About this chapter
Vectors in mechanics describes motion in two dimensions with i pointing east and j pointing north. Ships, boats, pucks and aircraft move with constant velocity, and the questions ask where they are at a given time, whether they collide, and how close they get.
This chapter covers magnitude and direction, bearings, adding vectors, parallel vectors, speed and velocity, position at time t, relative position, conditions such as being due north of a point, and the least distance between two ships found by completing the square.
The ten questions
- Q1 (6 marks): a particle with velocity 4i − 3j: its bearing, its position at time t, and when it is due east of O.
- Q2 (6 marks): a hiker's two legs on given bearings: the distance and bearing back to the start.
- Q3 (6 marks): two displacements, an angle with i, and a resultant parallel to 2i − j.
- Q4 (6 marks): a ship's velocity from its speed and direction, and its position at a clock time.
- Q5 (6 marks): two particles' positions at time t, the vector between them, and the distance.
- Q6 (10 marks): two ships: the velocity of one from a meeting condition, and their positions at clock times.
- Q7 (10 marks): a boat changes direction partway: its speed, its position later, and its distance from a lighthouse.
- Q8 (10 marks): a puck's velocity from two positions, where it was at t = 0, and a direction condition.
- Q9 (10 marks): an aircraft's velocity plus a wind: the speed and bearing over the ground.
- Q10 (11 marks): two ships that pass through the same point at different times, and their least distance apart.
Key skills tested
Magnitude and direction. The magnitude of ai + bj is √(a² + b²). For the angle, draw the vector and use tan θ = b/a on the right-angled triangle.
Bearings. Measure clockwise from north, as three figures. 4i − 3j points south of east, on a bearing of 90° + 36.9° = 127°.
Adding vectors. Add the i and j parts separately, or draw a triangle with the vectors top to tail and use the sine and cosine rules.
Parallel vectors. Parallel to 2i − j means a multiple of it, so for xi + yj the ratio x/2 must equal y/(−1).
Speed and velocity. Speed is the magnitude of the velocity. Speed 26 in the direction 5i − 12j gives a velocity of 2(5i − 12j), since 5i − 12j has length 13.
Position at time t. Position = starting position + velocity × t, where the start is t = 0. From t = 1 to t = 9 is 8 seconds, not 9.
Relative position. The vector from A to B is B's position minus A's. Particles collide only if both components agree at the same t.
Direction conditions. Due north of O means the i part is 0. North-east of O means the i and j parts are equal and positive.
Least distance. The distance squared is a quadratic in t. Complete the square, or differentiate, to find the least value and when it happens.
Worked example
Question 1 from this chapter. A particle P moves with constant velocity (4i − 3j) m/s, and at t = 0 it is at (−10i + 18j) m. (a) Find its direction as a bearing. (b) Write down its position at time t. (c) Find when P is due east of O, and its distance from O then.
(a) The velocity points 3 south for every 4 east, which is 36.9° below east. The bearing is 90° + 36.9° = 127°.
(b) r = (−10 + 4t)i + (18 − 3t)j.
(c) Due east means the j part is 0: 18 − 3t = 0, so t = 6. Then r = 14i, so P is 14 m from O.
A quick sketch of the direction stops the common slip of giving the acute angle 36.9° as the bearing.
Seen on real papers
Every question in our booklets is original. These are the recent papers where each question type has appeared.
- January 2025, Q5: a velocity found between two times, then the position at another time and a distance, as in Q8.
- January 2025, Q1(b): a resultant parallel to a given vector fixes an unknown, as in Q3.
Where marks are lost
- Counting the time wrongly. From t = 1 to t = 9 is 8 seconds. In January 2025 most spotted this; using 9 gives the wrong velocity.
- The position at a later time. Finding where the ball was at t = 7 was the harder part in January 2025. It needs a known position plus the velocity times the time elapsed since then.
- No vectors. Some tried a non-vector method on a vector question and could not score.
- Equating to the vector itself. For a resultant parallel to 7i + 2j, some set it equal to 7i + 2j instead of a multiple of it, and got the wrong unknowns.
Common questions
How do I turn a velocity into a bearing?
Sketch the vector from a north line, find the angle with tan, then measure clockwise from north.
What is the difference between speed and velocity?
Velocity is a vector with a direction; speed is its magnitude, always positive.
How do I show two ships collide?
Find a single value of t at which both the i and the j components of their positions are equal.
Where this chapter leads
- Mechanics 1 Chapter 4: Dynamics of a Particle Moving in a Straight Line: forces as vectors, and F = ma in i and j form.
- Mechanics 1 Chapter 7: Statics of a Particle: forces in components that add to zero.