3.11.2 Intersections of Lines & Planes

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

• A line is parallel to a plane if its direction vector is perpendicular to the plane’s normal vector
• If you know the Cartesian equation of the plane in the form  then the values of a, b, and c are the individual components of a normal vector to the plane
• The scalar product can be used to check in the direction vector and the normal vector are perpendicular
  • If two vectors are perpendicular their scalar product will be zero

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

• If the line is parallel to the plane then it will either never intersect or it will lie inside the plane
  • Check to see if they have a common point
• If a line is parallel to a plane and they share any point, then the line lies inside the plane

How do I find the point of intersection of a line and a plane?

• If a line is not parallel to a plane it will intersect it at a single point
• If both the vector equation of the line and the Cartesian equation of the plane is known then this can be found by:
• STEP 1: Set the position vector of the point you are looking for to have the individual components x, y, and z and substitute into the vector equation of the line
  • 
• STEP 2: Find the parametric equations in terms of x, y, and z
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  • 
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• STEP 3: Substitute these parametric equations into the Cartesian equation of the plane and solve to find λ
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• STEP 4: Substitute this value of λ back into the vector equation of the line and use it to find the position vector of the point of intersection
• STEP 5: Check this value in the Cartesian equation of the plane to make sure you have the correct answer

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
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    • 
    • 
  • 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