Intersection of 3 Planes
1 Introduction

The intersection of three planes in can result in several geometric configurations. We solve the system of three linear equations in three variables,

The possible outcomes are:

  1. Unique Point: The three planes intersect at a single common point. (Normals are not coplanar).
  2. Line of Intersection: The three planes intersect along a common line. (This happens if, for example, one plane contains the line of intersection of the other two).

  3. No Intersection:

    1. All three planes are parallel and distinct.
    2. Two planes are parallel and distinct, intersected by a third.
    3. The planes intersect in pairs forming three parallel lines (a "triangular prism" configuration).
    4. Two planes are coincident, and the third is parallel and distinct.
  4. Plane of Intersection: All three planes are coincident. (Or two are coincident, and the third intersects them in their common plane).

The method is to use elimination (as in 9.2) to reduce the 3x3 system.

  • If (where ) arises, there is no solution.
  • If arises, there are infinitely many solutions (either a line or a plane).
  • If a unique is found, that's the single point of intersection.
1.1 Example : Unique Point of Intersection

This was solved as an Example:

Solution

1.2 Example: Intersection in a Line

Find the intersection of the planes:

Solution

Step 1: Eliminate using (1) and (2). Add (1) and (2):

Step 2: Eliminate using (1) and (3).

Multiply (1) by 5:

Add (1') and (3):

Step 3: Analyze system (4) and (5).

These equations are identical—which means we effectively have only two distinct equations (e.g., (1) and (4), or (2) and (4)).
This signifies an infinite number of solutions, forming a line.

Let . From (4),

Substitute and into one of the original plane equations (e.g., (1)) to find :

The line of intersection is:

or

Check normals (optional but insightful):

Are they coplanar?

Check scalar triple product:

Since the scalar triple product is zero, the normal vectors are coplanar. This is consistent with intersection in a line (or parallel configurations, but we found a line).

1.3 Example : No Common Intersection (Triangular Prism)

Determine the intersection of:

Solution

Step 1: Eliminate using (1) and (2). Subtract (1) from (2):

Step 2: System of (3) and (4).

Subtract (3) from (4):

This is a contradiction. Therefore, the system has no solution.

Geometric Interpretation:

Let's check the normals:

No two normals are parallel. So no two planes are parallel.

  • and intersect in a line .
  • and intersect in a line .
  • and intersect in a line .

The fact that (a contradiction) arose from combining information from all three planes means these three lines of intersection ( ) must be parallel and distinct, forming a triangular prism shape. There is no point common to all three planes.

1.4 Example : All Three Planes Parallel and Distinct

Describe the intersection of:

Solution

The normal vector for all three planes is

So all three planes are parallel.

Since the values ( ) are different and not proportional in a way that would make them coincident (e.g. is ), the planes are distinct.

Thus, there is no intersection.

Attempting to solve this system, e.g., subtracting from :

This contradiction confirms no solution.

2 Homework
2.1 Question 4.1 (Easy)

Find the point of intersection of the following three planes:

Solution

We need to solve the system of three linear equations:

Step 1: Eliminate using (1) and (2).
Add (1) and (2):

Step 2: Eliminate using (1) and (3).
This requires careful choice or modification. Let's use (2) and (3).
Add (2) and (3):

(5)

Step 3: Solve the 2x2 system (4) and (5).

From (5), . Substitute this into (4):

Substitute into :

Step 4: Substitute and into an original equation (e.g., (1)) to find .

Answer: The point of intersection is .

2.2 Question 4.2 (Medium)

Analyze the intersection of the following three planes. Describe the geometric configuration.

Solution

Observe and :

The normal vector for is .
The normal vector for is .
We see that , so and are parallel.
If we multiply by 2, we get .
Comparing this with , the constant terms are different ( ).
Thus, and are parallel and distinct.

The third plane is . Its normal vector is not parallel to or . So is not parallel to or .
Since and are parallel and distinct, they never intersect. Therefore, there can be no point common to all three planes. will intersect in a line, and will intersect in another line, and these two lines of intersection will be parallel.

Geometric Configuration: Two planes ( ) are parallel and distinct. The third plane ( ) intersects both of these planes, creating two parallel lines of intersection. There is no common point of intersection for all three planes.

Algebraic verification: Try to solve the system:

Multiply (1) by 2: (1')
Subtract (1') from (2):

This is a contradiction.
Therefore, the system has no solution.

Answer: The system has no solution. Geometrically, planes and are parallel and distinct, and plane intersects them in two parallel lines.

2.3 Question 4.3 (Harder)

Consider the planes:

The planes and intersect in a line . Determine the relationship that must exist between and such that the plane also contains the line (i.e., the three planes intersect in the line ).

Solution

  • Step 1: Find the parametric equation of the line of intersection of and .

    Add (1) and (2) to eliminate and :

    This is interesting; is constant. This means the line of intersection is in a plane parallel to the -plane.
    Substitute into (1):

    Let (parameter).
    Then

    .
    So, the line of intersection is .
    In vector form: .
    A point on is (when ).
    The direction vector of is .

Step 2: For to contain the line , two conditions must be met:

  • The normal vector of , , must be perpendicular to the direction vector of , .

  • Any point on line must also be on plane .
    Let's use . Substitute into :

So, the conditions are and .
We can express in terms of fewer parameters. Let and .
Then and .
The equation of would be .
This is .

Verification by substituting parametric equations of L into Pi3:

Substitute

For this to be true for all (meaning the line is in the plane), we need:

  • Coefficient of to be zero: .
  • Constant terms to be equal: .

These are the same conditions found earlier.

Answer: The relationship between must be:

Any plane of the form (where are not both zero to define a plane) will contain the line .

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