Planes Passing Through the Intersection of Two Planes (Cartesian Form)

In 3D geometry, understanding the relationship between planes is crucial for solving various problems. One interesting scenario is when we want to find the equation of a plane that passes through the line of intersection of two given planes. This topic is particularly important for students preparing for exams that include 3D geometry problems.

Understanding the Basics

Before diving into the topic, let's establish some basic concepts:

• A plane in 3D space can be defined by a linear equation of the form (Ax + By + Cz + D = 0), where (A), (B), (C), and (D) are real numbers, and (x), (y), and (z) are the coordinates of any point on the plane.
• The line of intersection of two planes occurs where they meet in 3D space. If two planes are not parallel, they will intersect in a line.

Cartesian Form of a Plane

The Cartesian form of a plane's equation is given by:

[ Ax + By + Cz + D = 0 ]

Here, (A), (B), and (C) are the direction ratios of the normal to the plane, and (D) is the distance from the origin to the plane along the normal, with the appropriate sign.

Intersection of Two Planes

Consider two planes with equations:

[ \text{Plane 1: } A_1x + B_1y + C_1z + D_1 = 0 ] [ \text{Plane 2: } A_2x + B_2y + C_2z + D_2 = 0 ]

The line of intersection of these two planes can be found by solving these two equations simultaneously.

Plane Passing Through the Intersection of Two Planes

The equation of a plane passing through the intersection of Plane 1 and Plane 2 can be written as:

[ P(x, y, z) = k_1(A_1x + B_1y + C_1z + D_1) + k_2(A_2x + B_2y + C_2z + D_2) = 0 ]

where (k_1) and (k_2) are real numbers, not both zero. This is known as the family of planes passing through the line of intersection of Plane 1 and Plane 2.

Table of Differences and Important Points

Aspect	Intersection of Two Planes	Plane Passing Through Intersection
Definition	The common line where two planes meet.	A plane that contains the line of intersection of two other planes.
Equation	Obtained by solving the equations of the two planes simultaneously.	A linear combination of the equations of the two intersecting planes.
Parameters	Defined by the coefficients of the plane equations.	Defined by the coefficients of the plane equations and the parameters (k_1) and (k_2).
Uniqueness	The intersection line is unique if the planes are not parallel.	There are infinitely many planes that can pass through the intersection line.

Formulas

• Equation of a plane in Cartesian form: (Ax + By + Cz + D = 0)
• Equation of a family of planes: (P(x, y, z) = k_1(A_1x + B_1y + C_1z + D_1) + k_2(A_2x + B_2y + C_2z + D_2) = 0)

Examples

Example 1: Finding the Family of Planes

Given two planes:

[ \text{Plane 1: } x + 2y + 3z - 4 = 0 ] [ \text{Plane 2: } 2x - y + z + 5 = 0 ]

Find the family of planes passing through their intersection.

Solution:

The family of planes is given by:

[ P(x, y, z) = k_1(x + 2y + 3z - 4) + k_2(2x - y + z + 5) = 0 ]

This is the general equation of the plane passing through the intersection of Plane 1 and Plane 2.

Example 2: Specific Plane Passing Through Intersection

Using the same planes from Example 1, find a specific plane that passes through their intersection and is perpendicular to the plane (x + y + z = 0).

Solution:

The normal to the plane (x + y + z = 0) is (<1, 1, 1>). We want our plane to have the same normal, so we set (k_1 + 2k_2 = 1), (2k_1 - k_2 = 1), and (3k_1 + k_2 = 1). Solving these equations gives us (k_1) and (k_2). Substituting these values into the family of planes equation will give us the specific plane we are looking for.

Conclusion

Understanding how to find the equation of a plane passing through the intersection of two planes is a fundamental skill in 3D geometry. By mastering the concept of the family of planes and the ability to manipulate parameters, students can solve a wide range of problems related to the positioning and intersection of planes in space.