Vector addition, subtraction and scalar multiples
Use directed displacements and components to combine or reverse vectors.
Extended only. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Read column-vector components.
- Use signed arithmetic.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is BA in terms of AB?
What you will learn
- Add and subtract vectors component by component.
- Interpret scalar multiplication and reversed direction.
Addition follows a head-to-tail journey
For a = (3,−1) and b = (−2,4), a + b = (1,3). Move along a then b and the resultant joins the start to the final position. Swapping the journey order has the same total displacement, even though the intermediate point differs.
A negative vector reverses its direction: −b = (2,−4). Subtraction a − b means a + (−b), giving (5,−5). A directed vector AB points from A to B; BA = −AB. Direction matters, unlike an ordinary undirected length.
Scalars multiply both components
For scalar k, ka multiplies both horizontal and vertical components. Thus 2a = (6,−2), while −2a = (−6,2). Positive scalars preserve direction, negative scalars reverse it, and the magnitude is scaled by |k|. A zero scalar produces the zero vector.
Use arrows or bold lower-case letters on paper to distinguish vectors from numbers. In this text, a and b refer to directed vectors. Expressions such as 2a − b can be simplified either through components or as an algebraic vector expression; only add quantities with the same vector meaning.
Pause and explain
a = (3,−1), b = (−2,4). Find a + b.
Worked example
a = (3,−1), b = (−2,4). Find 2a − b.
Show the worked solution
- 2a = (6,−2).
- Subtract b component by component: (6−(−2),−2−4).
- The result is (8,−6).
Answer (8,−6).
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 (2,5) + (−4,1).
Give me a hint
Follow each component's axis.
Compare my reasoning
- Horizontal sum is 2 − 4 = −2.
- Vertical sum is 5 + 1 = 6.
- The resultant is (−2,6).
(−2,6).
Look for these in your work
- Added components.
- Preserved negative signs.
For a = (2,−3), b = (4,1), find 3a − 2b.
Give me a hint
Multiply before subtracting.
Compare my reasoning
- 3a = (6,−9), 2b = (8,2).
- Subtract to get (−2,−11).
- Check by adding 2b back to recover 3a.
(−2,−11).
Look for these in your work
- Scaled both components.
- Checked the subtraction.
A journey has displacements (5,2), (−1,4) and a final return to its start. Find the return vector.
Give me a hint
The entire closed journey sums to zero.
Compare my reasoning
- The first two total (4,6).
- The return must be the negative resultant.
- Use (−4,−6); all three sum to (0,0).
(−4,−6).
Look for these in your work
- Recognised the closed-path condition.
- Reversed the resultant.
Common mistakes
- Adding a horizontal to a vertical component.
- Subtracting without reversing the second vector.
- Treating directed AB as BA.
What can you explain now?
What does a negative scalar do to a nonzero vector?
Compare with the explanation
Reverses direction and scales magnitude by its absolute value.
The sign controls direction while the absolute factor controls physical length.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, draw a head-to-tail journey and calculate its return vector.
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.