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Determine parainetric cquations for the line L representing the interscction of the planes x – y + 2z = 0 and 3r + y – z = 1. %3D

Determine parainetric cquations for the line L representing the interscction of the planes x – y + 2z = 0 and 3r + y – z = 1. %3D

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Problem Statement:** Determine parametric equations for the line \( L \) representing the intersection of the planes \( x - y + 2z = 0 \) and \( 3x + y - z = 1 \). **Explanation:** To find the line of intersection between two planes, we need to follow these steps: 1. **Equations of the Planes:** - Plane 1: \( x - y + 2z = 0 \) - Plane 2: \( 3x + y - z = 1 \) 2. **Finding a Direction Vector for the Line:** - The direction vector of the line can be found by taking the cross product of the normal vectors of the two planes. - Normal vector of Plane 1: \( \mathbf{n}_1 = (1, -1, 2) \) - Normal vector of Plane 2: \( \mathbf{n}_2 = (3, 1, -1) \) - Direction vector \( \mathbf{d} = \mathbf{n}_1 \times \mathbf{n}_2 \) 3. **Finding a Point on the Line:** - Solve the system of equations represented by the two planes to find a common point \( (x, y, z) \). 4. **Writing the Parametric Equations:** - Use the point found and the direction vector to write the parametric equations for the line \( L \). This problem has no graphs or diagrams to explain further.
Transcribed Image Text:**Problem Statement:** Determine parametric equations for the line \( L \) representing the intersection of the planes \( x - y + 2z = 0 \) and \( 3x + y - z = 1 \). **Explanation:** To find the line of intersection between two planes, we need to follow these steps: 1. **Equations of the Planes:** - Plane 1: \( x - y + 2z = 0 \) - Plane 2: \( 3x + y - z = 1 \) 2. **Finding a Direction Vector for the Line:** - The direction vector of the line can be found by taking the cross product of the normal vectors of the two planes. - Normal vector of Plane 1: \( \mathbf{n}_1 = (1, -1, 2) \) - Normal vector of Plane 2: \( \mathbf{n}_2 = (3, 1, -1) \) - Direction vector \( \mathbf{d} = \mathbf{n}_1 \times \mathbf{n}_2 \) 3. **Finding a Point on the Line:** - Solve the system of equations represented by the two planes to find a common point \( (x, y, z) \). 4. **Writing the Parametric Equations:** - Use the point found and the direction vector to write the parametric equations for the line \( L \). This problem has no graphs or diagrams to explain further.
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