Subtract a vector by adding its opposite
Reverse both components for an additive inverse · Subtract matching components · Interpret the geometric difference arrow
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
- Vector addition
- Opposite directions
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
- Reverse both components for an additive inverse
- Subtract matching components
- Interpret the geometric difference arrow
Add the opposite
v−w means v+(−w). The vector −w has the same magnitude as w and opposite direction unless w is zero. Negate both components, not just one. For v = [4; 1], w = [1; 3], the difference is [3; −2].
Difference between two endpoints
If two arrows v and w share a tail, the arrow from the head of w to the head of v represents v−w. Reversing that endpoint order gives w−v = −(v−w). Subtraction is not generally commutative; the equality v−w = w−v holds only when the vectors are equal.
Sources for this lesson
Pause and explain
If v and w share a tail, which arrow represents v−w?
Worked example
For v = [4; 1], w = [1; 3], find v−w and w−v.
Show the worked solution
- v−w = [4−1; 1−3] = [3; −2].
- w−v = [−3; 2], the opposite vector.
Answer [3; −2] and [−3; 2], respectively.
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.
Find [5; 2]−[2; −1].
Give me a hint
Subtract both signed components.
Compare my reasoning
- 5−2 = 3.
- 2−(−1) = 3.
[3; 3].
Look for these in your work
- Keep the subtraction order
- Subtract a negative correctly
For u = [−2; 4], find −u and u−u.
Give me a hint
Negate both components for the opposite.
Compare my reasoning
- −u = [2; −4].
- u−u = [0; 0].
[2; −4]; [0; 0].
Look for these in your work
- Reverse both components
- Recognise cancellation
Two movements from O end at A with vector [2; 5] and B with vector [6; 1]. Find →AB and its reverse.
Give me a hint
The arrow from A to B is endpoint B minus endpoint A.
Compare my reasoning
- →AB = [6−2; 1−5] = [4; −4].
- →BA = [−4; 4].
→AB = [4; −4]; →BA = [−4; 4].
Look for these in your work
- Identify the endpoint order
- Check that the reverse is opposite
Common mistakes
- Subtract the signed second component correctly.
- The order of subtraction changes the direction.
What can you explain now?
If v and w share a tail, which arrow represents v−w?
Compare with the explanation
From the head of w to the head of v.
Its endpoint difference is v minus w.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
If v and w share a tail, which arrow represents v−w?
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.