What is the Equation of a Plane Containing Two Intersecting Lines?

S7-SA2-0280

Grade Level:

Class 12

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

Definition

What is it?

The equation of a plane containing two intersecting lines is a mathematical expression that describes the flat surface on which both lines lie. It's like finding a single sheet of paper that perfectly covers two pens that cross each other on a table.

Simple Example

Quick Example

Imagine two roads in your city, like MG Road and Brigade Road, crossing at a signal. If you wanted to build a flat, transparent roof over just that intersection, the equation of the plane would describe the exact position and tilt of that roof. The roof would contain both roads.

Worked Example

Step-by-Step

Let's find the equation of a plane containing two intersecting lines L1: (x-1)/2 = (y-2)/3 = (z-3)/4 and L2: (x-1)/3 = (y-2)/4 = (z-3)/5.

Step 1: Identify the point of intersection. Both lines pass through the point (1, 2, 3). This is our common point A(x1, y1, z1) = (1, 2, 3).
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Step 2: Identify the direction vectors of the lines. For L1, the direction vector is b1 = (2, 3, 4). For L2, the direction vector is b2 = (3, 4, 5).
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Step 3: Calculate the normal vector 'n' to the plane by taking the cross product of the direction vectors b1 and b2.
n = b1 x b2 = ( (3*5 - 4*4)i - (2*5 - 4*3)j + (2*4 - 3*3)k )
n = (15 - 16)i - (10 - 12)j + (8 - 9)k
n = -1i + 2j - 1k. So, the normal vector is (-1, 2, -1).
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Step 4: Use the general equation of a plane: A(x - x1) + B(y - y1) + C(z - z1) = 0, where (A, B, C) are the components of the normal vector and (x1, y1, z1) is the point on the plane.
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Step 5: Substitute the values: -1(x - 1) + 2(y - 2) - 1(z - 3) = 0.
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Step 6: Simplify the equation: -x + 1 + 2y - 4 - z + 3 = 0.
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Step 7: Combine terms: -x + 2y - z = 0. Or, multiplying by -1, x - 2y + z = 0.
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Answer: The equation of the plane is x - 2y + z = 0.

Why It Matters

Understanding planes and lines is crucial in fields like AI/ML for creating 3D models, in Engineering for designing structures, and in Physics for analyzing motion. For example, self-driving cars use these concepts to map their surroundings and navigate safely, creating a 'mental' 3D model of the road.

Common Mistakes

MISTAKE: Using the direction vector of only one line instead of both to find the normal vector. | CORRECTION: The normal vector must be perpendicular to BOTH direction vectors of the lines, so you must use the cross product of both direction vectors.

MISTAKE: Not ensuring the lines actually intersect before trying to find the plane. | CORRECTION: First, check if the given lines intersect. If they are parallel or skew (don't intersect), a single plane cannot contain both in the way described.

MISTAKE: Incorrectly performing the cross product calculation. | CORRECTION: Double-check each component (i, j, k) of the cross product carefully, paying attention to the signs.

Practice Questions

Try It Yourself

QUESTION: Find the equation of the plane containing the lines (x-0)/1 = (y-0)/2 = (z-0)/3 and (x-0)/2 = (y-0)/3 = (z-0)/1. | ANSWER: x - 5y + 7z = 0

QUESTION: Two lines intersect at the point (2, 1, 0). Their direction vectors are d1 = (1, 0, -1) and d2 = (0, 1, 2). What is the equation of the plane containing these lines? | ANSWER: x - 2y + z = 0

QUESTION: A plane contains the point (1, 1, 1) and is parallel to the vectors (2, -1, 3) and (1, 2, -1). Is this the same method as finding a plane containing two intersecting lines? Explain. | ANSWER: Yes, it uses the same method. The two vectors can be considered as the direction vectors of two intersecting lines (or lines passing through the point (1,1,1) with these directions). The cross product of these vectors gives the normal to the plane, and the point (1,1,1) is used in the plane equation.

MCQ

Quick Quiz

Which of these is the first step to find the equation of a plane containing two intersecting lines?

Find the distance between the two lines

Find the point of intersection of the two lines

Calculate the angle between the two lines

Determine if the lines are parallel

The Correct Answer Is:

To define a plane, you need a point on it and a normal vector. The point of intersection is a common point to both lines and thus lies on the plane. Options A, C, and D are not the direct first steps for this specific problem.

Real World Connection

In the Real World

In film-making and animation, especially in Bollywood, creating realistic 3D scenes often involves placing virtual cameras and objects in a virtual space. Understanding the equations of planes helps animators ensure that a character's walk on a virtual floor, or a camera's movement along a set path, looks natural and is mathematically correct.

Key Vocabulary

Key Terms

Plane: A flat, two-dimensional surface that extends infinitely in all directions. | Intersecting Lines: Two lines that cross each other at exactly one point. | Direction Vector: A vector that shows the direction of a line. | Normal Vector: A vector that is perpendicular (at 90 degrees) to a plane. | Cross Product: An operation on two vectors that results in a third vector perpendicular to both.

What's Next

What to Learn Next

Next, you can explore 'Equation of a Plane Passing Through Three Non-Collinear Points'. This builds on your understanding of how points define a plane, using similar vector concepts but with a different starting point.