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E. Find an equation of the plane passing through the point (6, 3, 5) and containing the line of intersection of the planes 2x−y+z=3 and 3x+y=z=12. Use vector operations.

E. Find an equation of the plane passing through the point (6, 3, 5) and containing the line of intersection of the planes 2x−y+z=3 and 3x+y=z=12. Use vector operations.

Holt Mcdougal Larson Pre-algebra: Student Edition 2012
Holt Mcdougal Larson Pre-algebra: Student Edition 2012
1st Edition
ISBN:9780547587776
Author:HOLT MCDOUGAL
Publisher:HOLT MCDOUGAL
Chapter1: Variables, Expressions, And Integers
Section1.8: The Coordinate Plane
Problem 7C
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**Problem 2**: Find an equation of the plane passing through the point (6, 3, 5) and containing the line of intersection of the planes \(2x - y + z = 3\) and \(3x + y - z = 12\). Use vector operations. Explanation: To solve this problem, you need to follow these steps: 1. **Line of Intersection of Two Planes**: The line of intersection of two planes is determined by finding the normal vectors of the planes and computing their cross-product. 2. **Normal Vectors**: The normal vector of the plane \(2x - y + z = 3\) is \(\mathbf{n_1} = (2, -1, 1)\), and the normal vector of the plane \(3x + y - z = 12\) is \(\mathbf{n_2} = (3, 1, -1)\). 3. **Cross Product**: Compute the cross product of \(\mathbf{n_1}\) and \(\mathbf{n_2}\) to find the direction vector \(\mathbf{d}\) of the line of intersection. 4. **Equation of the Plane**: Use the point (6, 3, 5) and the direction vector \(\mathbf{d}\) to establish the equation of the new plane. These steps involve vector operations such as cross-product, dot-product, and using the point-normal form of a plane equation. By following these steps, you can determine the equation of the required plane in vector and scalar forms.
Transcribed Image Text:**Problem 2**: Find an equation of the plane passing through the point (6, 3, 5) and containing the line of intersection of the planes \(2x - y + z = 3\) and \(3x + y - z = 12\). Use vector operations. Explanation: To solve this problem, you need to follow these steps: 1. **Line of Intersection of Two Planes**: The line of intersection of two planes is determined by finding the normal vectors of the planes and computing their cross-product. 2. **Normal Vectors**: The normal vector of the plane \(2x - y + z = 3\) is \(\mathbf{n_1} = (2, -1, 1)\), and the normal vector of the plane \(3x + y - z = 12\) is \(\mathbf{n_2} = (3, 1, -1)\). 3. **Cross Product**: Compute the cross product of \(\mathbf{n_1}\) and \(\mathbf{n_2}\) to find the direction vector \(\mathbf{d}\) of the line of intersection. 4. **Equation of the Plane**: Use the point (6, 3, 5) and the direction vector \(\mathbf{d}\) to establish the equation of the new plane. These steps involve vector operations such as cross-product, dot-product, and using the point-normal form of a plane equation. By following these steps, you can determine the equation of the required plane in vector and scalar forms.
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