Vector magnitude and unit directions
Calculate vector length from perpendicular components and recognise that equal lengths do not imply equal vectors.
Section Paper 4 · Extended · Calculator
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
- Use Pythagoras.
- Interpret signed vector components.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Can a vector magnitude be negative?
What you will learn
- Use |a| = √(x² + y²).
- Compare vector direction as well as magnitude.
Components form a right triangle
The magnitude of vector (x,y) is √(x² + y²). A vector (−6,8) therefore has length √(36 + 64) = 10 units. Signs indicate direction but disappear in the squared length. Magnitude is a non-negative scalar, not another column vector.
Two vectors can have equal magnitude while differing in direction: (3,4) and (−3,4) both have length five, but they are not equal. Equal vectors require both equal direction and equal magnitude, which means equal components in the same coordinate system.
Scaling length preserves the square-root relationship
The magnitude of ka is |k| times the magnitude of a. If a = (3,4), then −2a = (−6,−8) has magnitude ten. Do not multiply magnitude by a negative number and call the result a negative length.
An optional useful direction representation is a/|a| for a nonzero vector, which has magnitude one. For (3,4), this is (3/5,4/5). The zero vector cannot be normalised by division because its magnitude is zero. The syllabus requires magnitude; this unit-direction idea explains scaling rather than introducing another required formula.
Pause and explain
Find the magnitude of (−6,8).
Worked example
Find the magnitude of (−6,8).
Show the worked solution
- Square the perpendicular components: 36 and 64.
- Add them to obtain one hundred.
- Take the positive square root: ten units.
Answer 10 units.
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 |(5,12)|.
Give me a hint
Apply Pythagoras to the components.
Compare my reasoning
- 5² + 12² = 169.
- √169 = 13.
- The vector magnitude is thirteen units.
13 units.
Look for these in your work
- Squared before adding.
- Took the square root.
Find the magnitude of (−2,−3) exactly.
Give me a hint
Keep the final square root exact.
Compare my reasoning
- The component squares are four and nine.
- Their sum is thirteen.
- The magnitude is √13 units, regardless of both negative signs.
√13 units.
Look for these in your work
- Used both components.
- Kept an exact magnitude.
Vectors (3,4) and (4,3) both have length five. Are they equal? Explain.
Give me a hint
Equality requires the same displacement direction.
Compare my reasoning
- Both squared magnitudes equal twenty-five.
- Their horizontal and vertical components differ.
- They are not equal vectors even though their magnitudes are equal.
No; equal magnitude, different direction and components.
Look for these in your work
- Compared components.
- Distinguished vector from scalar equality.
Common mistakes
- Adding components instead of their squares.
- Reporting squared length.
- Equating vectors solely by magnitude.
What can you explain now?
What is the magnitude of −3a if |a| = 4?
Compare with the explanation
12.
Magnitude uses the absolute scale factor three; the minus sign reverses direction.
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
Tomorrow, give two unequal vectors of the same length and justify their difference.
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.