Add vectors by components and a parallelogram
Add matching components · Translate an arrow for head-to-tail addition · Distinguish the sum's magnitude from a sum of magnitudes
Exam scope and assessment notes
Extended content; not a Core requirement. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Components and magnitude
- Signed arithmetic
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Add matching components
- Translate an arrow for head-to-tail addition
- Distinguish the sum's magnitude from a sum of magnitudes
Two representations of one sum
For a = [a₁; a₂] and b = [b₁; b₂], a+b = [a₁+b₁; a₂+b₂]. Draw a, then place an unchanged copy of b at its head; the sum arrow joins the first tail to the last head. With both tails at the same point, the parallelogram diagonal gives the same sum.
Changing an arrow's starting location does not change its vector. Addition is commutative, so exchanging the head-to-tail order gives the same endpoint.
Straight displacement and path length
The diagram uses a = [3; 1], b = [1; 2], giving a+b = [4; 3] and magnitude 5. Their separate lengths total √10+√5, which exceeds 5. Magnitudes do not generally add when the directions differ.
Equal same-direction displacements can have additive magnitudes; do not replace the general warning by an always-unequal claim. For [3; 0] then [0; 4], path length is 7 but resultant displacement length is 5.
Sources for this lesson
Pause and explain
Must |a+b| equal |a|+|b|?
Worked example
Add [3; 1] and [1; 2] and calculate the sum's magnitude.
Show the worked solution
- Component sum is [4; 3].
- Its length is √(16+9) = 5, not √10+√5.
Answer Sum [4; 3]; magnitude 5.
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.
Add [2; 1] and [−1; 3].
Give me a hint
Add each pair of matching components.
Compare my reasoning
- Horizontal: 2−1 = 1.
- Vertical: 1+3 = 4.
[1; 4].
Look for these in your work
- Match component positions
- Check signs
Add [−3; 4] and [2; −6], then find the exact resultant magnitude.
Give me a hint
Calculate the vector before its length.
Compare my reasoning
- Sum = [−1; −2].
- Magnitude = √(1+4) = √5.
[−1; −2]; magnitude √5.
Look for these in your work
- Keep direction in the sum
- Take the length afterwards
A walker moves 3 m east then 4 m north. Compare total distance walked with the displacement vector and its length.
Give me a hint
A path and its straight resultant are different.
Compare my reasoning
- Distance walked is 3+4 = 7 m.
- Displacement is [3; 4] m, whose length is 5 m.
Distance 7 m; displacement [3; 4] m, magnitude 5 m.
Look for these in your work
- Identify the requested quantity
- Use the endpoint-to-endpoint displacement
Common mistakes
- Add horizontal to horizontal and vertical to vertical.
- The magnitudes of a and b are not generally additive.
What can you explain now?
Must |a+b| equal |a|+|b|?
Compare with the explanation
No; different directions can give a shorter resultant.
For perpendicular lengths 3 and 4, the resultant is 5 rather than 7.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Must |a+b| equal |a|+|b|?
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.