Mechanics · 26 September 2026 · 6 min read
M1 Mechanics vectors: components, position and motion questions
The symbols i and j do not introduce a new physical law. They organise perpendicular components so that motion and forces can be handled consistently. Most vector errors come from switching between a vector and its magnitude or comparing positions measured at different times.
This guide is for Pearson Edexcel International A-Level Mathematics Mechanics 1, WME01. The original examples focus on its component and modelling skills. Dr Desouky’s M1 tuition is online; M1 should not be confused with an IAL Physics paper simply because both courses use mechanics.
Name the vector and its units before calculating
A position vector describes where a particle is relative to an origin. A displacement is a change in position. Velocity describes the rate of position change, and acceleration the rate of velocity change. The same numerical pair can represent different physical quantities, so include units and a clear symbol rather than writing only “3i + 4j”.
Choose axes once and keep them throughout the problem. If i points east and j north, a negative j component means south. A negative component is not a negative speed. When a question asks for speed, you need the magnitude of velocity; when it asks for velocity, a scalar magnitude alone is incomplete.
Worked example: magnitude and direction of a force
A force is F = (6i + 8j) N. Its magnitude is √(6² + 8²) = 10 N. The angle measured anticlockwise from the positive i direction is tan⁻¹(8/6) = 53.1°. The direction statement matters: “53.1°” alone does not identify the reference axis or quadrant.
Now add a second force (−2i + 3j) N. The resultant is (4i + 11j) N, not a vector of magnitude 10 plus the second magnitude. An additional force required for equilibrium is (−4i − 11j) N. Add components first; only then find a magnitude if the question asks for it.
Worked example: position after constant-velocity motion
A particle starts at position (2i − j) m and moves with constant velocity (3i + 4j) m s⁻¹. After t seconds its position is r = (2 + 3t)i + (−1 + 4t)j metres. At t = 3 s it is at (11i + 11j) m. Its displacement is (9i + 12j) m, which differs from the final position vector.
The displacement magnitude is 15 m. Because this example has constant velocity along one straight line, the distance travelled is also 15 m. That equality does not hold for every journey. A changing path can have a distance larger than the magnitude of its net displacement, so do not turn this example into a universal rule.
Worked example: do two particles meet?
Keep particle A from the previous example. Particle B has position rB = (14 − t)i + 11j metres at the same time t. Equality of horizontal coordinates gives 2 + 3t = 14 − t, so t = 3 s. At that time A’s vertical coordinate is −1 + 4 × 3 = 11 m, matching B. Both coordinates agree, so the particles meet at (11, 11) m.
If B’s fixed vertical coordinate were 10 m instead, the horizontal equation would still give t = 3 s, but the particles would not meet. Crossing the same vertical line is not a collision. Two trajectories can also cross at a location reached at different times; position equality must use the same time variable.
Connect acceleration and force component by component
A 2.0 kg particle changes velocity from (i + 2j) to (7i − j) m s⁻¹ in 3.0 s under constant acceleration. The change is (6i − 3j) m s⁻¹, giving a = (2i − j) m s⁻². The resultant force is ma = (4i − 2j) N, with magnitude √20 = 4.47 N.
The negative vertical acceleration does not by itself say the particle is moving downward at every instant. Acceleration describes how the vertical velocity changes. Distinguishing the two prevents errors in both vector questions and ordinary one-dimensional motion problems.
Use a component audit before marking the answer
Check that the i equation contains only i components and the j equation only j components. Check whether the requested answer is a vector, magnitude, direction, position or elapsed time. Then substitute any meeting time back into both original position equations. A quick sketch is useful for spotting a quadrant error even when the algebra looks tidy.
For practice, change one initial position and decide whether a meeting still occurs; reverse one force component and find the new equilibrium force. Bring the full component equations when asking for help. They reveal whether the difficulty lies in the physical interpretation, the signs or the algebra.
Questions, explained
Choose a question for a direct answer, then explore the explanation and supporting resources. Each answer has its own link to save or share.
What is the difference between position and displacement vectors?
Position locates a particle relative to a chosen origin. Displacement is final position minus initial position. The final position vector is only equal to displacement when the initial position is the origin. Include the time associated with each position before subtracting.
How do I find speed from a velocity vector?
Take the magnitude. For perpendicular components ai + bj, speed is √(a² + b²), with the velocity units retained. Do not add the component values or treat a negative component as a negative speed. Velocity itself still requires direction.
How do I show two particles meet in an M1 question?
Set their position vectors equal at the same time and check both components. Solving only the horizontal equation is insufficient. Substitute the proposed time into both original vectors and confirm the same position, with a time that lies within the stated motion interval.
How do I find a force that makes a particle stay in equilibrium?
Add the existing forces component by component, then take the negative of that resultant. Equilibrium requires zero resultant force in every direction. If asked for magnitude and direction, calculate them from the balancing-force vector after the component addition.
Is M1 vectors part of Physics or Mathematics?
WME01 Mechanics 1 is a Pearson Edexcel International A-Level Mathematics unit. Its physical models overlap with Physics, but its examination requirements belong to the Mathematics specification. Use M1 papers and guidance for that unit; the academy’s M1 support is online.