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IBMathsDPAnalysis & Approaches (AA)HLRevision Notes3. Geometry & TrigonometryVector PropertiesAngle Between Two Vectors & Perpendicular Vectors

Angle Between Two Vectors & Perpendicular Vectors (DP IB Analysis & Approaches (AA)): Revision Note

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DP Analysis & Approaches (AA): HLDP Applications & Interpretation (AI): HLDP Applications & Interpretation (AI): HL

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Angle Between Two Vectors

How do I find the angle between two vectors?

  • If two vectors with different directions are placed at the same starting position, they will form an angle between them

  • The two formulae for the scalar product can be used together to find this angle

    • begin mathsize 16px style cos space invisible function application theta blank equals blank fraction numerator v subscript 1 w subscript 1 plus blank v subscript 2 w subscript 2 plus blank v subscript 3 w subscript 3 over denominator open vertical bar bold italic v close vertical bar vertical line bold italic w vertical line end fraction end style

    • This is given in the formula booklet

  • To find the angle between two vectors:

    • Calculate the scalar product between them

    • Calculate the magnitude of each vector

    • Use the formula to find cos θ

    • Use inverse trig to find θ

Examiner Tips and Tricks

  • The formula for this is given in the formula booklet so you do not need to remember it but make sure that you can find it quickly and easily in your exam

Worked Example

Calculate the angle formed by the two vectors begin mathsize 16px style bold italic v equals blank open parentheses fraction numerator negative 1 over denominator table row 3 row 2 end table end fraction close parentheses blank end styleand bold italic w equals 3 bold i plus 4 bold j minus bold k.

3-9-4-ib-aa-hl-angle-between-two-vectors-we-solution-a

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Perpendicular Vectors

How do I know if two vectors are perpendicular?

  • If the scalar product of two (non-zero) vectors is zero then they are perpendicular

    • If bold italic v times bold italic w equals 0 then v and w must be perpendicular to each other

  • Two vectors are perpendicular if their scalar product is zero

    • The value of cos θ = 0 therefore |v||w|cos θ = 0

 

Worked Example

Find the value of t such that the two vectors begin mathsize 16px style bold italic v equals blank open parentheses fraction numerator 2 over denominator table row t row 5 end table end fraction close parentheses blank end styleand bold italic w equals left parenthesis t minus 1 right parenthesis bold i minus bold j plus bold k are perpendicular to each other.

3-9-4-ib-aa-hl-the-angle-between-vectors-we-solution

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    Amber

    Author: Amber

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    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.