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IBMathsDPAnalysis & Approaches (AA)HLRevision Notes3. Geometry & TrigonometryVector PlanesAngles Between a Line & a Plane

Angles Between a Line & a Plane (DP IB Analysis & Approaches (AA)): Revision Note

Amber

Written by: Amber

Reviewed by: Dan Finlay

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

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Angle Between Line & Plane

What is meant by the angle between a line and a plane?

  • The angle between a line and a plane is defined to be the angle between:

    • The line

    • The projection of the line onto the plane

      • This is the line of intersection between the plane and a perpendicular plane which contains the line

  • This is the smallest angle between the line and the plane

  • It is easiest to think of these two lines making a right-triangle with the normal vector to the plane

    • The line joining the plane will be the hypotenuse

    • The line on the plane will be adjacent to the angle

    • The normal will the opposite the angle

Diagram of a line intersecting a plane showing angles, with normal vector labels, a right angle, and notes about the direction vector and intersection points.
Example of the angles between a line and a plane

How do I find the angle between a line and a plane?

  • For example, consider:

    • The line with equation l space colon space bold italic r equals open parentheses table row 3 row cell negative 1 end cell row cell negative 2 end cell end table close parentheses plus lambda open parentheses table row 1 row cell negative 2 end cell row 0 end table close parentheses

    • The plane with equation capital pi space colon space 2 x plus 3 y minus z equals 5

  • STEP 1

    Find the acute angle between the direction vector of the line and the normal vector to the plane

    • Use the formula cos alpha equals fraction numerator vertical line bold italic b times bold italic n vertical line over denominator vertical line bold italic b vertical line vertical line bold italic n vertical line end fraction

      • cos alpha equals fraction numerator open vertical bar open parentheses table row 1 row cell negative 2 end cell row 0 end table close parentheses times open parentheses table row 2 row 3 row cell negative 1 end cell end table close parentheses close vertical bar over denominator open vertical bar open parentheses table row 1 row cell negative 2 end cell row 0 end table close parentheses close vertical bar open vertical bar open parentheses table row 2 row 3 row cell negative 1 end cell end table close parentheses close vertical bar end fraction equals fraction numerator 4 over denominator square root of 70 end fraction

      • alpha equals cos to the power of negative 1 end exponent open parentheses fraction numerator 4 over denominator square root of 70 end fraction close parentheses equals 61.439... degree

  • STEP 2
    Subtract this angle from 90° to find the acute angle between the line and the plane

    • Subtract the angle from straight pi over 2 if working in radians

    • theta equals 90 degree minus 61.439... degree equals 28.560... degree

Examiner Tips and Tricks

Remember that if the scalar product is negative, your answer will result in an obtuse angle. Therefore, taking the absolute value of the scalar product means that you always get the acute angle.

Worked Example

Find the angle in radians between the line L with vector equation bold italic r equals open parentheses 2 minus lambda close parentheses bold italic i plus open parentheses lambda plus 1 close parentheses bold italic j plus open parentheses 1 minus 2 lambda close parentheses bold italic k and the plane capital pi with Cartesian equation x minus 3 y plus 2 z equals 5.

3-11-3-ib-hl-aa-angle-line-and-plane-we-solution-1

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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.