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 in Cartesian form?

• 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
  • 
  • 
  • 
• STEP 3: Substitute these parametric equations into the Cartesian equation of the plane and solve to find λ
  • 
• 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 I find the point of intersection of a line and a plane in vector form?

• Suppose you have a line with equation ?and plane with equation
• Form three equations with unknowns t, λ and μ
• Solve them simultaneously on your calculator
• Substitute the values back in to get the intersection

How do I find the angle between a line and a plane?

• When you find the angle between a line and a plane you will be finding the angle between the line itself and the line on the plane that creates the smallest angle with it
  • This means the line on the plane directly under the line as it joins the plane
• It is easiest to think of these two lines making a right-triangle with the normal vector to the plane
  • The line joining the plane will be the?hypotenuse
  • The line on the plane will be?adjacent?to the angle
  • The normal will the?opposite?the angle
• As you do not know the angle of the line on the plane you can instead find the angle between the?normal?and the?hypotenuse
  • This is the angle?opposite?the angle you want to find
  • This angle?can be found?because you will know the direction vector of the line joining the plane and the normal vector to the plane
  • This angle is also equal to the angle made by the line at the point it joins the plane and the normal vector at this point
• The smallest angle between the line and the plane will be 90° minus the angle between the normal vector and the line
  • In radians this will be ?minus the angle between the normal vector and the line