Find the magnitude of a vector
Convert a two-dimensional vector into a length with Pythagoras and keep magnitude distinct from direction.
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
Which theorem gives the magnitude from perpendicular vector components?
What you will learn
- Calculate a vector's magnitude from its components
- Use magnitude to check a displacement or side length
- Simplify exact roots and apply units in context
The components are perpendicular lengths
For vector v = (x, y), its magnitude is |v| = √(x² + y²). The signs affect direction but disappear after squaring.
For displacement AB, calculate the vector first, then its magnitude. A negative component is not a negative distance; length is non-negative.
Keep length separate from vector notation
The vector (3, 4) describes a displacement; its magnitude is 5. Use units such as metres only when components represent measured lengths.
Simplify square roots by taking out square factors, for example √72 = 6√2. Check with Pythagoras or an approximate decimal if needed.
Pause and explain
Can a vector's magnitude be negative?
Worked example
Find the magnitude of vector (5, 12).
Show the worked solution
- Use |v| = √(x² + y²).
- |v| = √(5² + 12²) = √169.
- |v| = 13.
Answer 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.
Find |(8, −15)|.
Give me a hint
Square both components, add and take the positive square root.
Compare my reasoning
- 8² = 64 and (−15)² = 225.
- 64 + 225 = 289.
- Magnitude = √289 = 17.
17.
Look for these in your work
- Negative sign squared
- Squares added
- Non-negative length returned
A displacement is v = (−6, 8) metres. Find its magnitude.
Give me a hint
The signs give direction; the magnitude is the positive length.
Compare my reasoning
- |v| = √((−6)² + 8²).
- |v| = √(36 + 64) = √100.
- |v| = 10 m.
10 m.
Look for these in your work
- Components squared
- Positive root selected
- Metres included
A robot's displacement is (7, 5) units. Find its straight-line distance from the start and explain why it is shorter than the route along the axes.
Give me a hint
Use the vector magnitude for the direct path; compare with the horizontal-plus-vertical route.
Compare my reasoning
- Direct distance = √(7² + 5²) = √74 units.
- Axis-aligned route = 7 + 5 = 12 units.
- √74 ≈ 8.60, which is shorter than 12 because the diagonal is the hypotenuse of the right triangle.
√74 units (about 8.60); this is shorter than the 12-unit axis route.
Look for these in your work
- Magnitude formula used
- Route length compared
- Pythagorean geometry explained
Common mistakes
- Adding components instead of their squares
- Taking a negative square root
- Reporting the vector where a length was requested
- Keeping square units for a magnitude
What can you explain now?
Find the magnitude of (3, 4).
Compare with the explanation
5
√(3² + 4²) = 5.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
What does the magnitude keep from a vector, and what directional information does it discard?
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.