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IBMathsDPApplications & Interpretation (AI)HLRevision Notes3. Geometry & TrigonometryVector Equations of LinesAngle Between Two Lines

Angle Between Two Lines (DP IB Applications & Interpretation (AI)): Revision Note

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Angle Between Two Lines

How do we find the angle between two lines?

  • The angle between two lines is equal to the angle between their direction vectors

    • It can be found using the scalar product of their direction vectors

  • Given two lines in the form bold italic r equals bold italic a subscript 1 plus lambda bold italic b subscript 1 and bold italic r equals bold italic a subscript 2 plus lambda bold italic b subscript 2 use the formula

    • begin mathsize 16px style theta equals cos to the power of negative 1 end exponent invisible function application open parentheses fraction numerator bold italic b subscript 1 blank bullet blank bold italic b subscript 2 over denominator open vertical bar bold italic b subscript 1 close vertical bar open vertical bar blank bold italic b subscript 2 close vertical bar end fraction close parentheses end style

  • If you are given the equations of the lines in a different form or two points on a line you will need to find their direction vectors first

  • To find the angle ABC the vectors BA and BC would be used, both starting from the point B

  • The intersection of two lines will always create two angles, an acute one and an obtuse one

    • A positive scalar product will result in the acute angle and a negative scalar product will result in the obtuse angle

      • Using the absolute value of the scalar product will always result in the acute angle

Examiner Tips and Tricks

  • In your exam read the question carefully to see if you need to find the acute or obtuse angle

    • When revising, get into the practice of double checking at the end of a question whether your angle is acute or obtuse and whether this fits the question

Worked Example

Find the acute angle, in radians between the two lines defined by the equations:

begin mathsize 16px style l subscript 1 colon space space bold italic a equals open parentheses table row 2 row 0 row cell blank 3 blank end cell end table close parentheses plus lambda open parentheses table row 1 row cell negative 4 end cell row cell blank minus 3 blank end cell end table close parentheses end style and  begin mathsize 16px style l subscript 2 colon space space bold italic b equals open parentheses table row 1 row cell negative 4 end cell row 3 end table close parentheses plus mu open parentheses table row cell blank minus 3 blank end cell row 2 row 5 end table close parentheses end style

R_UZJlZ8_3-10-3-ib-aa-hl-angle-between-we-solution-2a

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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.