IBMathsDPAnalysis & Approaches (AA)HLRevision Notes3. Geometry & TrigonometryVector Equations of LinesAngle Between Two Lines

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

Amber

Written by: Amber

Reviewed by: Dan Finlay

Updated on

DP Analysis & Approaches (AA): HLDP Analysis & Approaches (AA): HL

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Angle between two lines

How do we find the angles between two lines?

  • The angles between two lines are equal to the angles between their direction vectors

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

  • The intersection of two lines will always create two pairs of equal angles

    • one pair are acute

    • the other are obtuse

Intersecting lines with angles theta and 180 minus theta, labelled as r equals a1 plus lambda b1 and r equals a2 plus mu b2 in blue and red text.
Angles between two lines

  • You can find one angle between 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 using theta equals cos to the power of negative 1 end exponent open parentheses fraction numerator bold italic b subscript 1 blank bold times 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

  • You can find the other angle by subtracting this from 180°

    • Or from pi if you are working in radians

  • 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

  • You can always find the acute angle first by taking the absolute value of the scalar product

    •  cos to the power of negative 1 end exponent open parentheses fraction numerator open vertical bar bold italic b subscript bold 1 blank times blank bold b subscript bold 2 close vertical bar 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

Worked Example

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

l subscript 1 colon space space bold italic r 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 and  l subscript 2 colon space space bold italic r 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

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

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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.

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