Identify and resolve vectors
Separate scalar quantities from vectors, reverse a vector for subtraction, and resolve a vector into perpendicular components.
AS Level · Cambridge International Physics 9702, 2025–2027 syllabus, Topic 1. This online sequence supports learning and practice; it does not replace supervised laboratory work, full timed assessment or human subject-teacher review.
Before you begin
- Read a number with its unit and use multiplication, division, powers and right-angled triangles.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which of these is a vector quantity?
What you will learn
- Distinguish scalar from vector quantities using direction
- Add and subtract coplanar vectors
- Resolve a vector into two perpendicular components
Direction distinguishes a vector
A scalar has magnitude only. Examples include mass, time, energy, distance, speed, temperature and charge. A vector has both magnitude and direction. Examples include displacement, velocity, acceleration, force and momentum.
Two vectors with equal magnitude can have different effects if their directions differ. A negative sign in a one-dimensional calculation records direction relative to a chosen positive axis; it does not mean a negative size.
Add arrows, then use perpendicular components
To subtract vector B from vector A, add the reversed vector −B: A − B = A + (−B). In a scale diagram, place arrows head-to-tail; the resultant runs from the first tail to the final head.
For a vector of magnitude V at angle θ measured anticlockwise from +x, its perpendicular components are Vx = V cos θ and Vy = V sin θ. Give each component a sign from its direction. The resultant magnitude follows from Pythagoras and its direction from an inverse tangent with the correct quadrant.
Pause and explain
A vector of magnitude V makes angle θ from +x. What is its x-component?
Worked example
A force is 10 N at 30° above the positive horizontal direction. Resolve it into horizontal and vertical components.
Show the worked solution
- The angle is measured from the horizontal, so the horizontal component is adjacent: 10 cos 30°.
- The vertical component is opposite: 10 sin 30°.
- Both components are positive because the vector points right and upwards.
Answer Fx ≈ 8.66 N and Fy = 5.0 N.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A student walks 3.0 m east and then 4.0 m north. Find the displacement magnitude and direction.
Give me a hint
Treat east and north as perpendicular components.
Compare my reasoning
- Resultant magnitude = √(3.0² + 4.0²).
- The angle north of east is tan⁻¹(4.0/3.0).
- State the direction relative to the east axis.
5.0 m at about 53° north of east.
Look for these in your work
- Perpendicular components combined
- Magnitude and direction both given
Vector A is 8 N east and vector B is 5 N east. Find A − B.
Give me a hint
Reverse B before adding it to A.
Compare my reasoning
- −B is 5 N west.
- Add the signed horizontal components: +8 + (−5).
- Interpret the positive sign as east.
3 N east.
Look for these in your work
- Subtracted vector reversed
- Signed components combined
- Direction interpreted
A 12 N force acts 40° below the negative x-axis. Give its perpendicular components.
Give me a hint
The direction is in the third quadrant, so both components are negative.
Compare my reasoning
- The reference angle to the x-axis is 40°.
- Horizontal magnitude = 12 cos 40°; vertical magnitude = 12 sin 40°.
- Apply negative signs to both components.
Fx ≈ −9.19 N and Fy ≈ −7.71 N.
Look for these in your work
- Component magnitudes calculated
- Third-quadrant signs correct
Common mistakes
- Adding vector magnitudes without checking their directions
- Keeping vector B's direction when calculating A − B
- Swapping sine and cosine when the angle is measured from +x
- Ignoring a negative component in another quadrant
What can you explain now?
Which of these is a vector quantity?
Compare with the explanation
Velocity
Velocity has magnitude and direction.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
How do direction and angle convention determine the signs and values of vector components?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.