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)
- For example, letting x = λ gives:
- STEP 2: Rearrange the two equations to bring λ to one side
- Equations (1) and (2) become
- (1)
- (2)
- Equations (1) and (2) become
- STEP 3: Solve the equations simultaneously to find the two variables in terms of λ
- 3(1) – (2) Gives
- Substituting this into (1) gives
- 3(1) – (2) 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
- The parametric 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 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.