Scalars and coplanar vectors
Key idea: A vector needs both magnitude and direction. Components let you replace a sloping vector by two signed perpendicular parts that are easier to combine.
Continue where you stopped
The core idea
H1 Physics 8867 · Lesson 3 of 3
Check your understandingBy the end of this lesson, you should be able to
- Distinguish scalars from vectors using examples.
- Add and subtract coplanar vectors graphically or by components.
- Resolve a vector into signed perpendicular components and recover its resultant.
Learn the idea
Big question: How can two directed quantities be combined without losing the directions that make them vectors?
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
- A negative component is a direction, not a negative magnitude.
- Components are not extra vectors acting in addition to the original.
- State the reference direction for every final angle.
Relationships to know
Rₓ = Σx-componentsRᵧ = Σy-componentsR = √(Rₓ² + Rᵧ²)
Follow 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.
Now try it with support
Practise with support
A student walks 5.0 m east and then 12.0 m north. Find the resultant displacement.
Hints
- Use components (+5.0, +12.0) m.
- Find the angle from east, not from north.
View the guided answer
Magnitude = √(5.0² + 12.0²) = 13.0 m. Direction = tan⁻¹(12/5) = 67.4° north of east.
Your turn
Practise independently
Resolve a 24 N force directed 35° west of north into east–west and north–south components, including signs.
Check your answer
Taking east and north as positive, Fx = −24 sin 35° = −13.8 N and Fy = 24 cos 35° = +19.7 N. The negative x-component means west.
Common mistakes and exam guidance
Watch out for
- Adding vector magnitudes while ignoring direction.
- Using cosine for the component opposite the stated angle.
In an exam
- Draw a small labelled sketch before choosing sine or cosine.
- Give a direction with the final magnitude; a vector answer is incomplete without it.
Put the ideas together
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.
View the marking points and model answer
Marking points
- Finds net horizontal component 4 N west.
- Uses vertical component 12 N north.
- Obtains magnitude 12.6 or 12.7 N.
- Gives direction 18.4° west of north or equivalent.
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.
Finish from memory
Three-question recap
How is A − B constructed graphically?
Check
Reverse B, then add the reversed vector to A head to tail.
What does a negative x-component mean?
Check
The component points opposite the chosen positive x-direction.
Why must a vector answer include an angle reference?
Check
An angle alone does not identify the direction from which it was measured.
Continue with the next resource in this course.
Course and syllabus information
- Course
- GCE A-Level H1 Physics
- Edition
- GCE A-Level H1 Physics 2027