Calculate with two-dimensional vectors
Represent displacements as column vectors, find AB from position vectors, and add, subtract or scale vectors.
Section 8 · Transformations and vectors
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What does the vector from A to B equal using position vectors a and b?
What you will learn
- Read and write column-vector notation
- Find a displacement vector between two points
- Add, subtract and multiply vectors by a scalar
A vector records direction and distance
The column vector (x, y) means x units horizontally and y units vertically. A translation by this vector adds those components to every point.
If position vectors are a = OA and b = OB, then AB = b − a. Subtract corresponding components; the order matters because AB and BA point in opposite directions.
Use component operations consistently
Add or subtract vectors component by component. A scalar multiplies both components, so 3(x, y) = (3x, 3y). A negative scalar reverses direction as well as changing length.
Check a journey by chaining vectors head to tail: if A to B is u and B to C is v, then A to C is u + v.
Pause and explain
What happens when vector (x, y) is multiplied by 3?
Worked example
A(2, −1) and B(7, 3). Find vector AB.
Show the worked solution
- Subtract the coordinates of A from B.
- Horizontal component = 7 − 2 = 5.
- Vertical component = 3 − (−1) = 4, so AB = (5, 4).
Answer AB = (5, 4).
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 (3, −2) + (−5, 7).
Give me a hint
Add the first components and then the second components.
Compare my reasoning
- Horizontal component: 3 + (−5) = −2.
- Vertical component: −2 + 7 = 5.
- The sum is (−2, 5).
(−2, 5).
Look for these in your work
- Components paired correctly
- Signs retained
- Column result given
Find 2u − v when u = (4, −1) and v = (3, 5).
Give me a hint
Scale u first, then subtract v component by component.
Compare my reasoning
- 2u = (8, −2).
- 2u − v = (8−3, −2−5).
- The result is (5, −7).
(5, −7).
Look for these in your work
- Scalar applied to both components
- Subtraction order preserved
- Signs checked
A delivery drone flies from A(1, 2) to B(6, 4), then to C(3, 10). Find AB, BC and the direct displacement AC. Verify the journey relation.
Give me a hint
Find each displacement by subtracting its start coordinates from its endpoint coordinates.
Compare my reasoning
- AB = (6−1, 4−2) = (5, 2).
- BC = (3−6, 10−4) = (−3, 6).
- AC = (3−1, 10−2) = (2, 8), and AB + BC = (5−3, 2+6) = (2, 8).
AB = (5, 2), BC = (−3, 6), AC = (2, 8) = AB + BC.
Look for these in your work
- Endpoint subtraction used
- Intermediate journey combined
- Direct displacement verified
Common mistakes
- Reversing the endpoints when finding AB
- Adding vector components in a diagonal-length question
- Multiplying only one component by a scalar
- Treating a displacement as an absolute position
What can you explain now?
If a = (2, −3) and b = (5, 4), find a + b.
Compare with the explanation
(7, 1)
Add horizontal components and vertical components separately.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
How does vector subtraction encode the direction from a starting point to an endpoint?
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.