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IBMaths: AI HLDPRevision Notes3. Geometry & Trigonometry3.7 Vector Properties3.7.3 Magnitude of a Vector

Magnitude of a Vector (DP IB Maths: AI HL)

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Magnitude of a Vector

How do you find the magnitude of a vector?

  • The magnitude of a vector tells us its size or length
    • For a displacement vector it tells us the distance between the two points
    • For a position vector it tells us the distance of the point from the origin
  • The magnitude of the vector AB with rightwards arrow on top is denoted open vertical bar AB with rightwards arrow on top close vertical bar
    • The magnitude of the vector a is denoted |a|
  • The magnitude of a vector can be found using Pythagoras’ Theroem
  • The magnitude of a vector bold italic v equals v subscript 1 bold i plus blank v subscript 2 bold j plus v subscript 3 bold k blank is found using
    • open vertical bar bold italic v close vertical bar equals blank square root of v subscript 1 superscript space 2 end superscript plus blank v subscript 2 superscript blank 2 end superscript plus v subscript 3 superscript blank 2 end superscript end root
    • where begin mathsize 16px style v equals blank open parentheses table row cell v subscript 1 over v subscript 2 end cell row cell v subscript 3 end cell end table close parentheses end style
    • This is given in the formula booklet

Vectors in 3 Dimensions Diagram 2a, AS & A Level Maths revision notes

How do I find the distance between two points?

  • Vectors can be used to find the distance (or displacement) between two points
    • It is the magnitude of the vector between them
  • Given the position vectors of two points:
    • Find the displacement vector between them
    • Find the magnitude of the displacement vector between them

Examiner Tip

  • Finding the magnitude of a vector is the same as finding the distance between two coordinates, it is a useful formula to commit to memory in order to save time in the exam, however it is in your formula booklet if you need it

Worked example

Find the magnitude of the vector AB = 4i – j + 2k.

3-9-3-ib-aa-hl-magnitude-we-solution-a

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Unit Vectors

What is a unit vector?

  • A unit vector has a magnitude of 1
  • It can be found by dividing a vector by its magnitude
    • This will result in a vector with a size of 1 unit in the direction of the original vector
  • A unit vector in the direction of a is denoted fraction numerator bold a over denominator open vertical bar bold a close vertical bar end fraction
    • For example a unit vector in the direction 3i – 4j is fraction numerator left parenthesis 3 bold i blank minus 4 bold j right parenthesis over denominator square root of 3 squared plus 4 squared end root end fraction equals 3 over 5 bold i minus 4 over 5 bold j blank

Examiner Tip

  • Finding the unit vector will not be a question on its own but will be a useful skill for further vectors problems so it is important to be confident with it

Worked example

Find the unit vector in the direction 2i – 2j + k.

3-9-3-ib-aa-hl-magnitude-we-solution-b

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    Amber

    Author: Amber

    Expertise: Maths

    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.