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Combinations of Planes (Edexcel A Level Further Maths): Revision Note

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Amber

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Intersection of Planes

How do we find the line of intersection of two planes?

  • Two planes will either be parallel or they will intersect along a line

    • Consider the point where a wall meets a floor or a ceiling

    • You will need to find the equation of the line of intersection

  • If you have the Cartesian forms of the two planes then the equation of the line of intersection can be found by solving the two equations simultaneously

    • As the solution is a vector equation of a line rather than a unique point you will see below how the equation of the line can be found by part solving the equations

    • For example:    

      • 2 x minus y plus 3 z equals 7                           (1)

      • x minus 3 y plus 4 z equals 11 blank                       (2)

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

    • For example, letting x = λ gives:

      • 2 straight lambda minus y plus 3 z equals 7 blank                          (1)

      • straight lambda minus 3 y plus 4 z equals 11 blank                        (2)

  • STEP 2: Rearrange the two equations to bring λ to one side

    • Equations (1) and (2) become

      •  y minus 3 z equals 2 lambda blank minus blank 7 blank                       (1)

      • 3 y minus 4 z equals lambda minus blank 11 blank                       (2)

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

    • 3(1) – (2) Gives

      • z blank equals blank 2 blank minus lambda blank

    • Substituting this into (1) gives

      • y equals blank minus 1 minus lambda

  • STEP 4: Write the three parametric equations for x, y, and z in terms of λ and convert into the vector equation of a line in the form begin mathsize 16px style open parentheses fraction numerator x over denominator table row y row z end table end fraction close parentheses equals blank open parentheses fraction numerator x subscript 0 over denominator table row cell y subscript 0 end cell row cell z subscript 0 end cell end table end fraction close parentheses plus lambda open parentheses fraction numerator l over denominator table row m row n end table end fraction close parentheses end style

    • The parametric equations

      • x equals lambda

      • y equals negative 1 space minus space lambda

      • z equals 2 space minus space lambda

    • Become

      • open parentheses fraction numerator x over denominator table row y row z end table end fraction close parentheses equals open parentheses fraction numerator 0 over denominator table row cell negative 1 end cell row 2 end table end fraction close parentheses plus lambda open parentheses fraction numerator 1 over denominator table row cell negative 1 end cell row cell negative 1 end cell end table end fraction close parentheses

  • If you have fractions in your direction vector you can change its magnitude by multiplying each one by their common denominator

    • The magnitude of the direction vector can be changed without changing the equation of a line

  • An alternative method is to find two points on both planes by setting either x, y, or z to zero and solving the system of equations using your calculator

    • Repeat this twice to get two points on both planes

    • These two points can then be used to find the vector equation of the line between them

    • This will be the line of intersection of the planes

    • This method relies on the line of intersection having points where the chosen variables are equal to zero

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

Angle between two Planes

How do we find the angle between two planes?

  • The angle between two planes is equal to the angle between their normal vectors

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

  • If two planes Π1 and Π2 with normal vectors n1 and n2 meet at an angle then the two planes and the two normal vectors will form a quadrilateral

    • The angles between the planes and the normal will both be 90°

    • The angle between the two planes and the angle opposite it (between the two normal vectors) will add up to 180°

3-11-3-ib-hl-aa-angle-between-two-planes-diagram-2

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 between the two planes which can be defined by equations capital pi subscript 1 colon blank 2 x minus y plus 3 z equals blank 7 and capital pi subscript 2 colon blank x plus 2 y minus z equals 20.

al-fm-6-2-2-angle-between-two-planes-we-solution

Combinations of three Planes

What are the possible configurations of three planes?

  • Form three equations using the three planes

    •  a subscript 1 x plus b subscript 1 y plus c subscript 1 z equals d subscript 1 a subscript 2 x plus b subscript 2 y plus c subscript 2 z equals d subscript 2 a subscript 3 x plus b subscript 3 y plus c subscript 3 z equals d subscript 3

  • Let the matrix M be equal to the coefficients

    •  M equals open square brackets table row cell a subscript 1 end cell cell b subscript 1 end cell cell c subscript 1 end cell row cell a subscript 2 end cell cell b subscript 2 end cell cell c subscript 2 end cell row cell a subscript 3 end cell cell b subscript 3 end cell cell c subscript 3 end cell end table close square brackets

  • If det M not equal to 0 then the three planes intersect at a single point

    • open parentheses table row x row y row z end table close parentheses equals M to the power of negative 1 end exponent open parentheses table row cell d subscript 1 end cell row cell d subscript 2 end cell row cell d subscript 3 end cell end table close parentheses

  • If det M equals 0 then the three planes could

    • Be coincident or parallel

      • Check if the normal vectors are parallel

      • If they are coincident then there will be infinitely many solutions

      • If they are parallel then there will be no solutions

    • Intersect at a line

      • This is configuration is called a sheaf

    • Form a triangular prism

      • This is where pairs of planes interest at lines which are parallel to each other

    • Two could be parallel and the third could interest each plane separately

6-2-3-edexcel-al-fm-combination-of-3-planes

How can I find the configuration of three planes?

  • If the matrix of coefficients is non-singular then the planes intersect at a single point

  • If the matrix is singular then check if any of the planes are parallel or coincident

    • 2 x plus 3 y plus 5 z equals 4 and 4 x plus 6 y plus 10 z equals 8are coincident as they are scalar multiples

    • 2 x plus 3 y plus 5 z equals 4 and 4 x plus 6 y plus 10 z equals 9are parallel as their normal vectors are parallel

  • If the planes are not parallel then try to check to see if the equations are consistent

    • Consistent equations will have solutions

    • Inconsistent equations will not have any solutions

  • If the planes are not parallel and the equations are consistent then they form a sheaf

    • They intersect at a line

    • Eliminating variables will lead to the equation of this line

    • Eliminating all variables will lead to a statement that is always true

      • Such as 0 = 0

  • If the planes are not parallel and the equations are inconsistent then they form a triangular prism

    • They do not intersect

    • Each pair of planes intersect a line and these three lines are parallel

    • Eliminating all variables will lead to a statement that is never true

      • Such as 0 = 1

Worked Example

Three planes have equations given by

table row cell x plus 2 y minus z end cell equals 3 row cell 3 x plus 7 y plus z end cell equals 4 row cell x minus 9 z end cell equals k end table

a) Given that the three planes intersect in a straight line, find the value of k.

al-fm-6-2-3-three-planes-we-solution-a

b) Find a vector equation for the line of intersection.

al-fm-6-2-3-three-planes-we-solution-b

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    Amber

    Author: Amber

    Expertise: Maths

    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.