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Cambridge IGCSE0580

Magnitude of a vector

Mathematics 0580 Chapter Notes

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Magnitude of a vector
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In this chapter

  1. 1What magnitude means
  2. 2Square, then add
  3. 3Exact and decimal answers
  4. 4Magnitude of a vector between two points
  5. 5Finding an unknown from a magnitude

1. What magnitude means

The magnitude of a vector is its size. It is a single number, not another vector.

The magnitude of a is written with modulus signs: |a|. The magnitude of overrightarrowAB is written |overrightarrowAB|.

The magnitude is found with Pythagoras’ theorem. The two components are the shorter sides of a right-angled triangle and the magnitude is the length of its longest side.

|dbinomxy| = √(x² + y²)

Examiner insight

“A number of candidates did not recall the meaning of magnitude and omitted this question part; others gave an answer as another vector, as co-ordinates, or as a negative number.” “Some appeared to not understand the modulus sign.”

Worked example 12 marks

Do not use a calculator. Find |a| when a = dbinom512.

  1. 1

    |a| = √(5² + 12²) = √(25 + 144) [1]

  2. 2

    = √(169) = 13 [1]

2. Square, then add

Square each component, then add the squares, then take the square root.

A negative component is squared inside brackets. (-4)² = 16 but -4² = -16.

The squares are added, never subtracted.

|dbinom-3-4| = √((-3)² + (-4)²)

Examiner insight

“Some were unable to square the negative terms when using Pythagoras’ theorem and omitted important brackets within their method.” “Giving an answer to 1 decimal place only and subtracting the squares of the components were two of the most common errors seen.”

Worked example 12 marks

Do not use a calculator. Find |dbinom-8-6|.

  1. 1

    √((-8)² + (-6)²) = √(64 + 36) [1]

  2. 2

    = √(100) = 10 [1]

3. Exact and decimal answers

When the sum is not a square number, leave the answer as a surd, or use a calculator and give a decimal correct to at least 3 significant figures. Write the unrounded value in your working before you round it.

Examiner insight

“A final answer in surd form was acceptable but decimals had to be given correct to at least 3 significant figures.” “A few gave an answer 5.4 without a more accurate value in working.”

Worked example 13 marks

Find |dbinom2-5|. Give your answer correct to 3 significant figures.

  1. 1

    √(2² + (-5)²) = √(4 + 25) [1]

  2. 2

    = √29 = 5.385… [1]

  3. 3

    = 5.39 [1]

4. Magnitude of a vector between two points

For overrightarrowAB, subtract the coordinates of A from those of B: end minus start. Then find the magnitude.

Worked example 13 marks

Do not use a calculator. A is the point (1, 2) and B is the point (4, 6). Find |overrightarrowAB|.

  1. 1

    overrightarrowAB = dbinom4 - 16 - 2 = dbinom34 [1]

  2. 2

    |overrightarrowAB| = √(3² + 4²) = √25 [1]

  3. 3

    = 5 [1]

5. Finding an unknown from a magnitude

Write the vector, form the equation, and square both sides. Take both the positive and the negative square root.

Worked example 14 marks

Point A has coordinates (-4, 3). Point C has coordinates (x, 7) and |overrightarrowAC| = 4√10. Find the two possible values of x.

  1. 1

    overrightarrowAC = dbinomx + 47 - 3 = dbinomx + 44 [1]

  2. 2

    (x + 4)² + 4² = (4√10)² = 160 [1]

  3. 3

    (x + 4)² = 160 - 16 = 144 [1]

  4. 4

    x + 4 = ± 12, so x = 8 or x = -16 [1]

Recap

  • The magnitude of a vector is its size. It is written with modulus signs, such as |a| or |overrightarrowAB|.
  • |dbinomxy| = √(x² + y²). Square, add, then square root.
  • Put brackets round a negative component: (-4)² = 16.
  • Give an exact surd, or a decimal to at least 3 significant figures with the unrounded value shown.
  • For two points, find the vector first: end minus start.
  • A magnitude equation can have two answers: take the positive and negative square root.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. Find |dbinom68|.2 marks
  2. Do not use a calculator. Find |dbinom5-12|.2 marks
  3. Do not use a calculator. Find |dbinom-7-24|.2 marks
  4. Find |dbinom-37|. Give your answer correct to 3 significant figures.2 marks
  5. Do not use a calculator. Find |dbinom2-2|. Give your answer as a surd in its simplest form.2 marks
  6. P is the point (-2, 1) and Q is the point (3, 7). Calculate |overrightarrowPQ|. Give your answer correct to 3 significant figures.3 marks
  7. A drone flies from the point (1, -4) to the point (-3, 2). Calculate the magnitude of the vector of this move. Give your answer correct to 3 significant figures.3 marks
  8. Do not use a calculator. The vector dbinomk12 has magnitude 13. Find the two possible values of k.3 marks
  9. A robot moves on a grid marked in metres. It travels in straight lines from A (4, -3) to B (-4, -1), then to C (1, 7), then back to A. Calculate the total distance it travels. Give your answer correct to 3 significant figures.5 marks
  10. Do not use a calculator. A is the point (k, 3) and B is the point (3k, 3 - k). |overrightarrowAB| = 6√5. Find the two possible values of k. For the positive value of k, find the coordinates of B.5 marks

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