# Edexcel A Level Further Maths: Core Pure:复习笔记6.2.3 Combinations of Planes

### 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 calculator
• 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

#### Worked Example

Two planes  and  are defined by the equations:

Find the vector equation of the line of intersection of the two planes.

### Angle between two Planes

#### How do we find the angle between two planes?

• The angle between two planes is equal to the angle between their normal vectors
• It can be found using the scalar product of their normal vectors
• If two planes Π1 and Π2 with normal vectors n1 and n2 meet at an angle then the two planes and the two normal vectors will form a quadrilateral
• The angles between the planes and the normal will both be 90°
• The angle between the two planes and the angle opposite it (between the two normal vectors) will add up to 180°

#### Exam Tip

• In your exam read the question carefully to see if you need to find the acute or obtuse angle
• When revising, get into the practice of double checking at the end of a question whether your angle is acute or obtuse and whether this fits the question

#### Worked Example

Find the acute angle between the two planes which can be defined by equations  and .

### Combinations of three Planes

#### What are the possible configurations of three planes?

• Form three equations using the three planes
• Let the matrix M be equal to the coefficients
• If  then the three planes intersect at a single point
• If  then the three planes could
• Be the coincident or parallel
• Check if the normal vectors are parallel
• If they are coincident then there will be infinitely many solutions
• If they are parallel then there will be no solutions
• Intersect at a line
• This is configuration is called a sheaf
• Form a triangular prism
• This is where pairs of planes interest at lines which are parallel to each other
• Two could be parallel and the third could interest each plane separately

#### How can I find the configuration of three planes?

• If the matrix of coefficients is non-singular then the planes intersect at a single point
• If the matrix is singular then check if any of the planes are parallel or coincident
• are coincident as they are scalar multiples
• are parallel as their normal vectors are parallel
• If the planes are not parallel then try to check to see if the equations are consistent
• Consistent equations will have solutions
• Inconsistent equations will not have any solutions
• If the planes are not parallel and the equations are consistent then they form a sheaf
• They intersect at a line
• Eliminating variables will lead to the equation of this line
• Eliminating all variables will lead to a statement that is always true
• Such as 0 = 0
• If the planes are not parallel and the equations are inconsistent then they form a triangular prism
• They do not intersect
• Each pair of planes intersect a line and these three lines are parallel
• Eliminating all variables will lead to a statement that is never true
• Such as 0 = 1

#### Worked Example

Three planes have equations given by

a) Given that the three planes intersect in a straight line, find the value of .

b) Find a vector equation for the line of intersection.