Add, subtract and scale vectors
Calculate vector sums, differences, scalar multiples and magnitudes component by component.
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
How do you add (a, b) and (c, d)?
What you will learn
- Add and subtract two-dimensional vectors
- Multiply a vector by a scalar
- Find a vector's magnitude and check the operation geometrically
Operate on matching components
For a = (a₁, a₂) and b = (b₁, b₂), a + b = (a₁ + b₁, a₂ + b₂), while a − b = (a₁ − b₁, a₂ − b₂). Keep the horizontal and vertical positions aligned.
A scalar k multiplies every component: ka = (ka₁, ka₂). A positive scalar stretches or shrinks without reversing direction; a negative scalar also reverses direction.
Magnitude measures length, not direction
The magnitude of (a, b) is √(a² + b²), by Pythagoras. It is a non-negative number, not a vector. The magnitude of −v equals the magnitude of v.
Use a right-triangle check when possible: if the components are 3 and 4, the magnitude is 5. Units for both components must match before combining physical vectors.
Pause and explain
What is the magnitude of (−6, 8)?
Worked example
Let a = (3, −2) and b = (−1, 4). Find a + b, a − b, 2a and |a|.
Show the worked solution
- Add and subtract matching components: a + b = (2, 2) and a − b = (4, −6).
- Scale both components: 2a = (6, −4).
- Find |a| = √(3² + (−2)²) = √13.
Answer a + b = (2, 2), a − b = (4, −6), 2a = (6, −4), |a| = √13.
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.
For u = (4, 1) and v = (−2, 5), find u + v, u − v and 3v.
Give me a hint
Apply each operation to horizontal and vertical components separately.
Compare my reasoning
- u + v = (4 − 2, 1 + 5) = (2, 6).
- u − v = (4 − (−2), 1 − 5) = (6, −4).
- 3v = (−6, 15).
u + v = (2, 6), u − v = (6, −4), 3v = (−6, 15).
Look for these in your work
- Component positions aligned
- Subtraction signs retained
- Both components scaled
Find |(5, 12)|, |−(5, 12)| and the vector ½(5, 12).
Give me a hint
Magnitude is unchanged by reversing direction.
Compare my reasoning
- The magnitude is √(25 + 144) = 13.
- The negative vector has the same magnitude 13.
- Half the vector is (5/2, 6).
Magnitudes 13 and 13; half-vector (5/2, 6).
Look for these in your work
- Pythagorean length correct
- Reversal does not change length
- Scalar applied to both components
A wind velocity is (−3, 4) m/s and a boat's velocity relative to the water is (5, 0) m/s. Find the resultant velocity and its speed.
Give me a hint
Add component velocities; then find the resultant magnitude.
Compare my reasoning
- The east-west component is −3 + 5 = 2 m/s.
- The north-south component is 4 + 0 = 4 m/s.
- The speed is √(2² + 4²) = √20 = 2√5 m/s.
Resultant velocity (2, 4) m/s; speed 2√5 m/s.
Look for these in your work
- Velocity components added
- Resultant direction sign retained
- Speed calculated with units
Common mistakes
- Adding unlike positions or units
- Scaling only one component
- Giving a magnitude a direction sign
- Squaring a negative component without its brackets
What can you explain now?
What is −2(1, −3)?
Compare with the explanation
(−2, 6)
Multiply both components by −2.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
How do vector operations work component by component, and how does magnitude differ from a 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.