Equation of Plane Through Intersection of Two Planes | Class 12

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What is the Equation of a Plane Passing Through the Intersection of Two Planes?

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

When two planes cross each other, they form a line where they meet. The equation of a plane passing through this line of intersection represents any plane that contains this common line. It's like finding all the pages that can be opened from the spine of a book.

Simple Example

Quick Example

Imagine two walls in your room meeting at a corner. That corner line is their intersection. Any flat surface (like a thin cardboard sheet) that touches this corner line from top to bottom is a plane passing through the intersection of those two walls. There can be many such sheets.

Worked Example

Step-by-Step

Let's find the equation of a plane passing through the intersection of planes P1: x + 2y + 3z - 4 = 0 and P2: 2x + y - z + 5 = 0. Also, this new plane passes through the point (1, 1, 1).

1. The general equation of a plane passing through the intersection of P1 and P2 is P1 + lambda * P2 = 0.
---2. Substitute the equations of P1 and P2: (x + 2y + 3z - 4) + lambda * (2x + y - z + 5) = 0.
---3. Since this new plane passes through the point (1, 1, 1), substitute x=1, y=1, z=1 into the equation:
(1 + 2(1) + 3(1) - 4) + lambda * (2(1) + 1 - 1 + 5) = 0.
---4. Simplify the equation: (1 + 2 + 3 - 4) + lambda * (2 + 1 - 1 + 5) = 0.
---5. This gives: (2) + lambda * (7) = 0.
---6. Solve for lambda: 2 + 7*lambda = 0, so 7*lambda = -2, which means lambda = -2/7.
---7. Substitute the value of lambda back into the general equation:
(x + 2y + 3z - 4) + (-2/7) * (2x + y - z + 5) = 0.
---8. Multiply by 7 to clear the fraction: 7(x + 2y + 3z - 4) - 2(2x + y - z + 5) = 0.
7x + 14y + 21z - 28 - 4x - 2y + 2z - 10 = 0.
Combine like terms: (7x - 4x) + (14y - 2y) + (21z + 2z) + (-28 - 10) = 0.
Answer: The equation of the plane is 3x + 12y + 23z - 38 = 0.

Why It Matters

Understanding how planes intersect is crucial in many fields. In AI/ML, it helps define decision boundaries for classifying data, like separating pictures of dogs from cats. In engineering, it's used to design complex structures like bridges or aircraft parts, ensuring different components fit together perfectly. Even in medicine, doctors use similar math for 3D imaging to locate tumors precisely.

Common Mistakes

MISTAKE: Forgetting to put lambda with the second plane's entire equation, or putting it with both planes. | CORRECTION: The general form is P1 + lambda * P2 = 0. Lambda multiplies only the second plane's equation.

MISTAKE: Making calculation errors when substituting the point and solving for lambda. | CORRECTION: Double-check all additions, subtractions, and multiplications, especially with negative signs, when finding the value of lambda.

MISTAKE: Not distributing the lambda value (especially if it's a fraction) correctly to all terms of the second plane's equation. | CORRECTION: After finding lambda, make sure to multiply every term (x, y, z, and constant) in the second plane's equation by lambda before combining terms.

Practice Questions

Try It Yourself

QUESTION: Find the equation of the plane passing through the intersection of the planes x + y + z = 6 and 2x + 3y + 4z + 5 = 0, and also passing through the point (1, 1, 1). | ANSWER: 20x + 23y + 26z - 69 = 0

QUESTION: Determine the equation of the plane passing through the intersection of the planes 3x - y + 2z - 4 = 0 and x + y + z - 2 = 0, and containing the point (2, 2, 1). | ANSWER: 7x + y + 6z - 22 = 0

QUESTION: A plane passes through the intersection of planes P1: x + 2y - z = 1 and P2: 2x + y + z = 2. If this new plane is parallel to the x-axis, find its equation. (Hint: If parallel to x-axis, coefficient of x in its normal vector is 0). | ANSWER: y - 2z = 0

MCQ

Quick Quiz

The general equation of a plane passing through the intersection of two planes P1 = 0 and P2 = 0 is:

P1 * P2 = 0

P1 + P2 = 0

P1 + lambda * P2 = 0

lambda * P1 + P2 = 0

The Correct Answer Is:

The correct way to represent any plane passing through the intersection of two given planes P1 and P2 is P1 + lambda * P2 = 0, where lambda is a constant. Options A and B are incorrect forms, and option D is essentially the same as C, just with lambda on the other plane.

Real World Connection

In the Real World

Think about how Google Maps or Ola/Uber show you routes. The 3D model of roads and buildings in these apps uses planes and their intersections to define structures. Urban planners in cities like Bengaluru or Mumbai use similar principles to design flyovers, metro lines, and building layouts, ensuring different levels and sections intersect correctly without collisions, using mathematical models of planes.

Key Vocabulary

Key Terms

PLANE: A flat, two-dimensional surface that extends infinitely in all directions. | INTERSECTION: The line or point where two or more geometric figures meet. | EQUATION OF A PLANE: An algebraic expression that defines the position and orientation of a plane in 3D space. | LAMBDA (lambda): A scalar constant used as a multiplier in mathematical equations, often to represent a family of solutions.

What's Next

What to Learn Next

Great job understanding how to find planes through intersections! Next, you can explore the 'Equation of a Plane Passing Through Three Non-Collinear Points.' This will build on your knowledge of planes and help you define them in different ways, preparing you for more advanced 3D geometry problems.