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Intersections of Two Planes (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

Intersection of two planes

How can I tell if two planes are parallel?

  • Two planes are parallel if their normal vectors are scalar multiples

    • A normal vector for the plane bold italic r equals bold italic a plus lambda bold italic b plus mu bold italic c is bold italic n equals bold italic b cross times bold italic c

  • You can spot this easier if the equations are written in Cartesian form a x plus b y plus c z equals d

    • Planes are parallel if the left-hand side of the equations are scalar multiples

      • For example, 2 x plus 3 y plus z equals 5 and 4 x plus 6 y plus 2 z equals 9 are parallel

      • For example, 2 x plus 3 y plus z equals 5 and 4 x plus 6 y plus z equals 9 are not parallel

How do I find the line of intersection of two non-parallel planes?

  • Two non-parallel planes intersect at a line

    • This is different from two lines which intersect at a point

  • To find the line of intersection you need to write the equations of the planes in Cartesian form

  • For example, consider the planes with equations   

    • capital pi subscript 1 space colon space 2 x minus y plus 3 z equals 7

    • capital pi subscript 2 space colon space x minus 3 y plus 4 z equals 11

Using algebra

  • STEP 1
    Choose one variable and substitute this variable for lambda in both equations

    • Using x equals lambda

      • 2 straight lambda minus y plus 3 z equals 7

      • straight lambda minus 3 y plus 4 z equals 11

  • STEP 2
    Rearrange the two equations to isolate the lambda terms and the constant terms on one side

    •  y minus 3 z equals 2 lambda minus 7

    • 3 y minus 4 z equals lambda minus 11

  • STEP 3
    Solve the equations simultaneously to find the two variables in terms of lambda

    • z equals 2 minus lambda

    • y equals negative 1 minus lambda

      • If there are no solutions, then choose a different variable to be lambda

  • STEP 4
    Write the three parametric equations for x, y, and z in terms of lambda

    • x equals lambda

    • y equals negative 1 minus lambda

    • z equals 2 minus lambda

Examiner Tips and Tricks

The direction vector of the line should be perpendicular to the normal vectors of both planes. Therefore, to check your answer, you can find the vector product of the two normal vectors and check whether it is a scalar multiple of the direction vector of the line.

Using the normal vector

  • STEP 1
    Find a direction vector of the line by taking the vector product of the two normal vectors of the planes

    • open parentheses table row 2 row cell negative 1 end cell row 3 end table close parentheses cross times open parentheses table row 1 row cell negative 3 end cell row 4 end table close parentheses equals open parentheses table row 5 row cell negative 5 end cell row cell negative 5 end cell end table close parentheses

  • STEP 2
    Find a point that lies on both planes by setting one of x, y, and z equal to zero and solving the equations simultaneously to find the other two

    • Using z equals 0

      • 2 x minus y equals 7

      • x minus 3 y equals 11

      • x equals 2 and y equals negative 3

  • STEP 3
    Write the equation in the form bold italic r equals bold italic a plus lambda bold italic b where bold italic a is the position vector of the point on both planes and bold italic b is the direction vector

    • bold italic r equals open parentheses table row 2 row cell negative 3 end cell row 0 end table close parentheses plus lambda open parentheses table row 5 row cell negative 5 end cell row cell negative 5 end cell end table close parentheses

Examiner Tips and Tricks

You can multiply the direction vector by a scalar to make the components simplified. For example, you can multiply this one by 1 fifth to get bold italic r equals open parentheses table row 2 row cell negative 3 end cell row 0 end table close parentheses plus lambda open parentheses table row 1 row cell negative 1 end cell row cell negative 1 end cell end table close parentheses.

Worked Example

Two planes capital pi subscript 1 and capital pi subscript 2 are defined by the equations:

capital pi subscript 1 colon blank 3 x plus 4 y plus 2 z equals 7

capital pi subscript 2 colon blank x minus 2 y plus 3 z equals 5

Find the vector equation of the line of intersection of the two planes.

3-11-2-ib-aa-hl-intersect-two-planes-we-solution-2-fixed

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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.