Plane Definition

When we talk about planes in math, we are talking about specific surfaces that have very specific properties. In order to understand the intersection of two planes, let’s cover the basics of planes.In the table below, you will find the properties that any plane will have.

Property 1 A flat, two dimensional surface
Property 2 It can extend forever in both directions
Property 3 It has no thickness

 

A plane is two-dimensional, which means that it only has two directions. Take a look at these directions below.

plane_two_dimensional

 

Direction 1 Up or down
Direction 2 Left or right

 

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Plane Interactions

When you have a plane, you can have many different interactions within, or inside, that plane. In fact, a graph is an example of a plane, where you can have many different lines interacting with each other.

plane_interactions

 

Interaction 1 Going in x or y directions
Interaction 2 A point on a graph (x,y)
Interaction 3 A line on a graph

 

When you have more than one plane, you can have interactions between planes. These interactions are summarized in the table below.

 

Interaction 1 Intersecting The planes intersect
Interaction 2 Parallel The planes are parallel to each other
Interaction 3 Coincident The planes are the same

 

Take a look at these interactions in the images below.

interactions_of_planes

 

A B C
Intersecting Parallel Coincident

 

Equation of a Plane

While a plane may be a two-dimensional surface, the plane itself is located in a three dimensional space. This may not make sense now, so let’s use an example. Say you have a sheet of paper in real-life.

directions_vectors

 

A Sheet of paper Has two directions Up or down and left or right
B Paper in space Has three directions Up or down and left or right and tilt

 

If you stick a pencil inside of a piece of paper, you can see that while the surface of the plane only has two directions, the plane itself can be moved around in any direction by that pencil. You can think of this pencil as the third direction of the plane.

 

vector_planes
This means that any point on the plane can be found using the length of the pencil, and the two directions on the plane.

 

three_dimensional
So, the equation of a plane can be found using the point on the graph and the vector of the “tilt.” However, you can typically be given the equation of planes in the following form. 

Equation
Plane Ax+By+Cz = D

 

Intersection of Planes

Recall that intersecting planes mean that they cross each other. You should be careful when dealing with intersecting planes:

 

Perpendicular Intersection
Meaning Planes that intersect at a 90 degree angle Planes that cross each other at any angle

 

When you have two planes intersecting one another, you have a line that forms where they touch each other. Take a look below.

 

line_of_intersection
Definition
Line of intersection The line where two planes meet

 

As you can see, this line has a special name, called the line of intersection. In order to find where two planes meet, you have to find the equation of the line of intersection between the two planes.

 

System of Equations

In order to find the line of intersection, let’s take a look at an example of two planes. Let’s take a look at the equation of two planes, which are written in the table below.

 

Plane 1 Plane 2
Formula x + 4y - z = 6 x - 2y + z = 4

 

You can think of these two lines as a system of equations. A system of equations is defined in the table below.

 

Definition Notation Example
System of Equations Two or more linear equations that have a relationship The linear equations are written on top of one another x + 4y - z = 6

x - 2y + z = 4

 

Parametric Form

When you have any line in space, you can write these linear equations in two ways:

  • Parametric form
  • Symmetric form

 

The easiest way to find the line of intersection is to work with the parametric form. The parametric form is defined in the table below.

 

Definition Parameter
Parametric Form The formula is written in terms of a parameter. t

 

In order to find the parametric form, we can use the system of equations from the previous section as an example and follow the steps in the table below.

 

Step 1 Solve the equations for one variable, like x x + 4y - z = 6 -> x = 6 -4y + z

x - 2y + z = 4 -> x = 4 + 2y - z

Step 2 Because the two equations equal to x, we can set them equal to each other 6 -3y + z = 4 + 2y - z
Step 3 Solve the equation for one variable, like z 2 + 2z = 6y

2z = 6y - 2

z = 3y - 1

Step 4 Replace the equation for z into one of the original equations from step 1 x - 2y + z = 4 -> x = 4 + 2y - z

x = 4 + 2y - (3y - 1)

x = 4 + 2y - 3y + 1

x = 5 + - y

 

Finding the Line of Intersection

Now that we’ve followed the steps from the previous section, we have three equations.

 

Equation 1 x = 5 + - y
Equation 2 y = y
Equation 3 z = 3y - 1

 

The equation for y is simply y = y. This is clear, because of course any given value for y is equal to itself. We could have done this for any of the other values (x=x, z=z).

 

Since they are all written in terms of the y value, all we have to do to get the equation of the line of intersection is replace these y terms with the parameter t. This gives us the parametric form of the line.

 

Equation 1 x = 5 + - t
Equation 2 y = t
Equation 3 z = 3t - 1

 

Together, these equations will give you any point on the line of intersection P(x,y,z).

 

Symmetric Form

If you want to have the line of intersection in symmetric form, you need to first find the parametric form of the line. Next, follow the two steps below.

 

Step 1 Set all equations for x, y and z in terms of the t parameter
Step 2 Set all equations equal to each other

 

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Danica

Located in Prague and studying to become a Statistician, I enjoy reading, writing, and exploring new places.