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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

Amber

Written by: Amber

Reviewed by: Dan Finlay

Updated on

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

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Angle Between Two Vectors

What is meant by the angle between two vectors?

  • The angle between two vectors is the angle formed by drawing both vectors from the same starting point

Two vectors, V in blue and W in red, diverge from a point, forming an angle θ between them, shown with a curved line.
Example of the angle between two vectors

How do I find the angle between two vectors?

  • You can find the angle between two vectors by using the formula bold italic v times bold italic w equals open vertical bar bold italic v close vertical bar open vertical bar bold italic w close vertical bar cos space theta

  • This can be rearranged to give 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

    • begin mathsize 16px style bold italic v equals blank open parentheses fraction numerator v subscript 1 over denominator table row cell v subscript 2 end cell row cell v subscript 3 end cell end table end fraction close parentheses end style

    • begin mathsize 16px style bold italic w equals blank open parentheses fraction numerator w subscript 1 over denominator table row cell w subscript 2 end cell row cell w subscript 3 end cell end table end fraction close parentheses end style

Examiner Tips and Tricks

This is given in the formula booklet under the geometry and trigonometry section.

  • 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 space theta

    • Use inverse trig to find theta

  • The sign of the scalar product determines whether theta is acute or obtuse

    • It is acute if the scalar product is positive

    • It is obtuse if the scalar product is negative

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?

  • Two vectors are perpendicular if their scalar product is zero

    • This is because the angle between two perpendicular vectors is 90°

    • cos 90 degree equals 0 which means bold italic v times bold italic w equals 0

  • The converse is also true

    • If bold italic v times bold italic w equals 0 for non-zero vectors

    • Then bold italic v and bold italic w must be perpendicular

Examiner Tips and Tricks

You might get told that two vectors are perpendicular and have to find an unknown component. As soon as you read that two vectors are perpendicular, you should take the scalar product and set it equal to zero.

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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    Author
    Reviewer
    Amber

    Author: Amber

    Expertise: Maths Content Creator

    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.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.