Intersections of Planes (DP IB Analysis & Approaches (AA)): Revision Note

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:    

    •                            (1)

    •                        (2)

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

  • For example, letting x = λ gives:

    •                           (1)

    •                         (2)

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

  • Equations (1) and (2) become

    •                         (1)

    •                        (2)

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

  • 3(1) – (2) Gives

    • 

  • Substituting this into (1) gives

    • 

• 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 

  • The parametric equations

    • 

    • 

    • 

  • Become

    • 

• 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 GDC or row reduction

  • 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

How do we find the relationship between three planes?

• Three planes could either be parallel, intersect at one point, or intersect along a line

• If the three planes have a unique point of intersection this point can be found by using your GDC (or row reduction) to solve the three equations in their Cartesian form

  • Make sure you know how to use your GDC to solve a system of linear equations

  • Enter all three equations in for the three variables x, y, and z

  • Your GDC will give you the unique solution which will be the coordinates of the point of intersection

• If the three planes do not intersect at a unique point you will not be able to use your GDC to solve the equations

  • If there are no solutions to the system of Cartesian equations then there is no unique point of intersection

• If the three planes are all parallel their normal vectors will be parallel to each other

  • Show that the normal vectors all have equivalent direction vectors

  • These direction vectors may be scalar multiples of each other

• If the three planes have no point of intersection and are not all parallel they may have a relationship such as:

  • Each plane intersects two other planes such that they form a prism (none are parallel)

  • Two planes are parallel with the third plane intersecting each of them

  • Check the normal vectors to see if any two of the planes are parallel to decide which relationship they have

• If the three planes intersect along a line there will not be a unique solution to the three equations but there will be a vector equation of a line that will satisfy the three equations

• The system of equations will need to be solved by elimination or row reduction

  • Choose one variable to substitute for λ

  • Solve two of the equations simultaneously to find the other two variables in terms of λ

  • Write x, y, and z in terms of λ in the parametric form of the equation of the line and convert into the vector form of the equation of a line