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
• 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 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
    • 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