Edexcel A Level Further Maths: Core Pure:复习笔记6.2.2 Combinations of Lines & Planes

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 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

Worked Example

Find the point of intersection of the line  with the plane

Angle between a Line & a Plane

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

Exam Tip

• Remember that if the scalar product is negative your answer will result in an obtuse angle
• Taking the absolute value of the scalar product will ensure that you get the acute angle as your answer

Worked Example

Find the angle in radians between the line L with vector equation  and the plane  with Cartesian equation .