Read and write two-dimensional vector notation
Represent a directed movement with components, column vectors, i and j notation, and vectors between named points.
Section 13 · Vectors in two dimensions
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What does the vector (3, −5) describe?
What you will learn
- Use column, component and unit-vector notation correctly
- Find a directed vector from one point to another
- Distinguish a vector's direction from its magnitude
A vector records both direction and magnitude
In a coordinate plane, (a, b) means a units horizontally and b units vertically. It can also be written as a two-entry column vector or as ai + bj, where i is one unit in the positive x-direction and j is one unit in the positive y-direction.
The vector AB starts at A and ends at B. Subtract the coordinates of A from those of B: AB = (xB − xA, yB − yA). Reversing direction changes the sign of both components: BA = −AB.
Equal vectors have the same components
Two vectors are equal when their horizontal components match and their vertical components match. They can be drawn in different places but still represent the same displacement.
The zero vector (0, 0) has zero magnitude and no direction. It is not a unit vector and cannot be normalised to a direction.
Pause and explain
Which notation is equivalent to the column vector (2, 7)?
Worked example
Find vector PQ when P = (−2, 4) and Q = (3, 1). Write it in component and i, j notation.
Show the worked solution
- Subtract the start point from the end point: (3 − (−2), 1 − 4).
- This gives horizontal change 5 and vertical change −3.
- Translate the components into unit-vector notation.
Answer PQ = (5, −3) = 5i − 3j.
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 vector AB for A = (1, −2) and B = (5, 1).
Give me a hint
Subtract A's coordinates from B's in the same order.
Compare my reasoning
- Horizontal change is 5 − 1 = 4.
- Vertical change is 1 − (−2) = 3.
- Combine the components.
AB = (4, 3) = 4i + 3j.
Look for these in your work
- End minus start used
- Both components correct
- Notation matches the components
For A = (5, 2) and B = (−1, 6), find BA and AB.
Give me a hint
Calculate each direction separately; reversing direction negates the result.
Compare my reasoning
- BA = A − B = (5 − (−1), 2 − 6) = (6, −4).
- AB = B − A = (−1 − 5, 6 − 2) = (−6, 4).
- Check that AB = −BA.
BA = (6, −4); AB = (−6, 4).
Look for these in your work
- Both directions calculated
- Signs agree with endpoints
- Opposite-vector check passed
A delivery robot moves 6 metres east and 8 metres north. Write its displacement vector in column and i, j notation.
Give me a hint
Use east as positive x and north as positive y.
Compare my reasoning
- The horizontal component is 6.
- The vertical component is 8.
- Write the same ordered pair in both notations.
(6, 8) = 6i + 8j metres.
Look for these in your work
- Coordinate directions stated
- Components in correct order
- Equivalent notations given
Common mistakes
- Subtracting the end point from the start point
- Swapping horizontal and vertical components
- Thinking equal vectors must have the same starting point
- Calling the zero vector a unit vector
What can you explain now?
If AB = (4, −2), what is BA?
Compare with the explanation
(−4, 2)
Reversing a vector negates each component.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Make it stick
How do you find a directed vector between two points, and what changes when you reverse it?
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.