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Intersections of a Line & a Plane (DP IB Analysis & Approaches (AA)): Revision Note

Jamie Wood

Written by: Jamie Wood

Reviewed by: Dan Finlay

Updated on

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

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Intersection of Line & Plane

How do I tell if a line is parallel to a plane?

  • A line is parallel to a plane if any direction vector of the line is perpendicular to the any normal vector of the plane

  • They are parallel if bold italic b times open parentheses table row a row b row c end table close parentheses equals 0 where

    • The equation of the line is in the form bold italic r equals bold italic a plus lambda bold italic b

    • The equation of the plane is in the form a x plus b y plus c z equals d

How do I tell if the line lies inside the plane?

  • Suppose a line is parallel to a plane

    • They could never intersect

    • Or the line could lie in the plane

  • The line lies in a parallel plane if any point on the line is also on the plane

    • You can check whether the coordinates of the point with position vector bold italic a satisfy the equation of the plane

How do I find the point of intersection of a line and a plane which are not parallel?

Using the Cartesian equation of the plane

  • For example, consider

    • the line l space colon space bold italic r equals open parentheses table row 1 row 5 row 6 end table close parentheses plus lambda open parentheses table row cell negative 1 end cell row 2 row 1 end table close parentheses

    • the plane capital pi space colon space 5 x plus 2 y minus z equals 13

  • STEP 1
    Find the three equations using the parametric equations of the line

    • x equals 1 minus lambda

    • y equals 5 plus 2 lambda

    • z equals 6 plus lambda

  • STEP 2
    Substitute these parametric equations into the Cartesian equation of the plane

    • 5 open parentheses 1 minus lambda close parentheses plus 2 open parentheses 5 plus 2 lambda close parentheses minus open parentheses 6 plus lambda close parentheses equals 13

  • STEP 3
    Solve to find the value of lambda

    • lambda equals negative 2

  • STEP 4
    Substitute this value of lambda back into the parametric equations

    • x equals 1 minus open parentheses negative 2 close parentheses equals 3

    • y equals 5 plus 2 open parentheses negative 2 close parentheses equals 1

    • z equals 6 plus open parentheses negative 2 close parentheses equals 4

Examiner Tips and Tricks

Check your answer by substituting the coordinates into the Cartesian equation of the plane.

Using a vector equation of the plane

  • For example, consider

    • the line l space colon space bold italic r equals open parentheses table row 1 row 5 row 6 end table close parentheses plus lambda open parentheses table row cell negative 1 end cell row 2 row 1 end table close parentheses

    • the plane capital pi space colon space bold italic r equals open parentheses table row 5 row cell negative 2 end cell row 8 end table close parentheses plus mu open parentheses table row 0 row 1 row 2 end table close parentheses plus nu open parentheses table row 1 row cell negative 1 end cell row 3 end table close parentheses

  • STEP 1
    Set the two equations equal to each and form three equations

    • 1 minus lambda equals 5 plus nu rightwards double arrow negative 4 equals lambda plus nu

    • 5 plus 2 lambda equals negative 2 plus mu minus nu rightwards double arrow 7 equals negative 2 lambda plus mu minus nu

    • 6 plus lambda equals 8 plus 2 mu plus 3 nu rightwards double arrow negative 2 equals negative lambda plus 2 mu plus 3 nu

  • STEP 2
    Solve the equations simultaneously

    • lambda equals negative 2

    • mu equals 1

    • nu equals negative 2

  • STEP 3
    Substitute the values back into the equation of the line or the equation of the plane

    • open parentheses table row 1 row 5 row 6 end table close parentheses plus open parentheses negative 2 close parentheses open parentheses table row cell negative 1 end cell row 2 row 1 end table close parentheses equals open parentheses table row 3 row 1 row 4 end table close parentheses

    • open parentheses table row 5 row cell negative 2 end cell row 8 end table close parentheses plus open parentheses 1 close parentheses open parentheses table row 0 row 1 row 2 end table close parentheses plus open parentheses negative 2 close parentheses open parentheses table row 1 row cell negative 1 end cell row 3 end table close parentheses equals open parentheses table row 3 row 1 row 4 end table close parentheses

Examiner Tips and Tricks

This method can be quicker if you have a GDC. However, if this comes up on the nom-calculator paper, then you should first write the equation of the plane in Cartesian form and use the first method.

Worked Example

Find the point of intersection of the line begin mathsize 16px style r blank equals blank open parentheses table row 1 row cell negative 3 end cell row 2 end table close parentheses plus lambda open parentheses table row 2 row cell negative 1 end cell row cell negative 1 end cell end table close parentheses end style with the plane 3 x minus 4 y plus z equals 8.

3-11-2-ib-aa-hl-intersect-line-plane-we-solution

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    Author
    Reviewer
    Jamie Wood

    Author: Jamie Wood

    Expertise: Maths Content Creator

    Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.

    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.