Scalars and coplanar vector operations
Key idea: A complete H2 Physics lesson on SI quantities, estimation, errors, uncertainty and coplanar vectors.
Continue where you stopped
The core idea
Build the idea
Learn the idea
Big question: How do components make vector addition and subtraction reliable?
A scalar has magnitude only; a vector also has direction. Choose perpendicular positive axes, resolve every vector with signed sine or cosine components, combine corresponding components and only then recover magnitude and direction. Subtracting a vector means adding its reverse.
Decide whether direction belongs in the answer
A scalar has magnitude only: mass, time, temperature, distance, speed and energy are examples. A vector has both magnitude and direction: displacement, velocity, acceleration, force and momentum are examples. Changing direction changes a vector even if its magnitude stays constant.
Check your understanding: A car keeps a constant speed while turning. Is its velocity constant?
No. Velocity is a vector, so its changing direction means the velocity changes.
Subtraction means adding a reversed vector
To find A − B, reverse B and add the reversed vector to A. This definition works in a scale drawing and in components. It also explains relative velocity: velocity of P relative to Q is vP − vQ.
A component is a signed projection, not part of the magnitude left over after subtraction. Once axes are chosen, every vector in the problem must use the same positive directions.
Check your understanding: If A is 5 N east and B is 2 N east, what is A − B?
3 N east. Reversing B gives 2 N west, which is then added to 5 N east.
Use geometry and components as cross-checks
A head-to-tail or parallelogram drawing reveals the approximate direction of the answer. Components then give precision. If the calculated resultant points outside the region suggested by the sketch, a sign or angle has probably been mishandled.
Check your understanding: Can a resultant of two non-zero vectors be zero?
Yes, but only when the two vectors have equal magnitudes and opposite directions.
Key ideas to keep
- A component sign comes from direction, not from the calculator.
- The resultant magnitude is not normally the sum of magnitudes.
- State the final direction with an angle and reference direction.
See the reasoning
Worked example
Combine two forces at an obtuse angle
Question: An 8.0 N force acts east. A 6.0 N force acts at 120° anticlockwise from east. Find the resultant magnitude and direction.
Step 1: Choose axes and resolve
Why: The obtuse angle makes the second force’s horizontal component negative.
Working: F₁ = (8.0, 0) N; F₂ = (6 cos120°, 6 sin120°) = (−3.0, 5.20) N.
Step 2: Add like components
Why: Only components along the same axis can be added directly.
Working: R = (8.0 − 3.0, 0 + 5.20) = (5.0, 5.20) N.
Step 3: Recover magnitude and direction
Why: The perpendicular component triangle contains the complete resultant.
Working: |R| = √(5.0² + 5.20²) = 7.21 N; θ = tan⁻¹(5.20/5.0) = 46.1°.
Answer: The resultant is 7.2 N at 46° north of east.
Check: Its magnitude lies between the 2 N difference and 14 N sum, and its direction lies between the two original directions.
Another worked model
Question
Add 8 N east to 6 N north, then subtract the 6 N north vector from the 8 N east vector.
Check the worked solution
With east as +x and north as +y, the sum is (8, 6) N: magnitude 10 N at 36.9° north of east. The difference is (8, −6) N: magnitude 10 N at 36.9° south of east.
Use a hint if needed
Practise with support
Try this
Resolve a 20 m displacement at 30° west of north into east and north components.
Hint: Declare east +x and north +y before assigning signs.
Check your answer
East component = −20 sin 30° = −10 m. North component = 20 cos 30° = 17.3 m.
Now work without the hint
Practise independently
Your turn
Two coplanar forces are A = (−3, 4) N and B = (5, −2) N. Find A + B and A − B, giving component and magnitude forms.
Check your answer
A + B = (2, 2) N with magnitude 2.83 N. A − B = (−8, 6) N with magnitude 10.0 N.
Avoid these traps
Common mistakes
Common mistake
Vector magnitudes may be added or subtracted without directions.
What is wrong with this reasoning?
Show better thinking
Choose axes, resolve each vector with signs, combine corresponding components, then recover magnitude and direction.
Write for the examiner
Exam guidance
Draw a quick arrow sketch and declare the axes before writing component equations.
Exam-style practice [4 marks]
Three forces act on a point: 12 N north, 5 N east and 9 N west. Determine the resultant force, including direction.
Plan before you answer
- Combine the east–west components with signs.
- Keep the north component separate.
- Use Pythagoras and state the angle reference.
Mark your answer and compare the model
Marking points
Tick each point only if your answer states it clearly.
Model answer
Taking east and north as positive, Rₓ = 5 − 9 = −4 N and Rᵧ = 12 N. Hence R = √(4² + 12²) = 12.6 N. The angle west of north is tan⁻¹(4/12) = 18.4°, so the resultant is 12.6 N, 18.4° west of north.
Come back in three days
Check what stayed with you
Recall question 1
How is A − B constructed graphically?
Check the answer
Reverse B, then add the reversed vector to A head to tail.
Recall question 2
What does a negative x-component mean?
Check the answer
The component points opposite the chosen positive x-direction.
Recall question 3
Why must a vector answer include an angle reference?
Check the answer
An angle alone does not identify the direction from which it was measured.
Syllabus and review details
This lesson covers the listed H2 Physics 9478 outcomes.
- GCE A-Level H2 PhysicsTopic 1(h) / Topic 1(i) / Topic 1(j) · 2027Checked against the syllabus · partial topic coverageOfficial 9478 syllabus
Course and syllabus information
- Course
- GCE A-Level H2 Physics
- Edition
- GCE A-Level H2 Physics 2027