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32. Find the equation of a plane through the origin and per- pendicular to x – y+ z = 5 and 2x + y– 2z = 7.

32. Find the equation of a plane through the origin and per- pendicular to x – y+ z = 5 and 2x + y– 2z = 7.

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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**Mathematics Problem Set** 32. **Problem**: Find the equation of a plane through the origin and perpendicular to the lines given by the equations \(x - y + z = 5\) and \(2x + y - 2z = 7\). 33. **Problem**: Given the points \( P = (1, 2, 3) \), \( Q = (3, 5, 7) \), and \( R = (2, 5, 3) \), solve the following: (a) Determine a unit vector perpendicular to the plane containing points \( P \), \( Q \), and \( R \). (b) Calculate the angle between the vectors \( \mathbf{PQ} \) and \( \mathbf{PR} \). (c) Find the area of the triangle \( PQR \). (d) Compute the distance from point \( R \) to the line passing through points \( P \) and \( Q \).
Transcribed Image Text:**Mathematics Problem Set** 32. **Problem**: Find the equation of a plane through the origin and perpendicular to the lines given by the equations \(x - y + z = 5\) and \(2x + y - 2z = 7\). 33. **Problem**: Given the points \( P = (1, 2, 3) \), \( Q = (3, 5, 7) \), and \( R = (2, 5, 3) \), solve the following: (a) Determine a unit vector perpendicular to the plane containing points \( P \), \( Q \), and \( R \). (b) Calculate the angle between the vectors \( \mathbf{PQ} \) and \( \mathbf{PR} \). (c) Find the area of the triangle \( PQR \). (d) Compute the distance from point \( R \) to the line passing through points \( P \) and \( Q \).
**Question 28:** Find a vector parallel to the line of intersection of the two planes \(4x - 3y + 2z = 12\) and \(x + 5y - z = 25\). --- **Explanation for Educational Context:** In this question, we are tasked with finding a vector that runs parallel to the line where two planes intersect. Each plane is described by its own linear equation in three-dimensional space: - The first plane is represented by the equation \(4x - 3y + 2z = 12\). - The second plane is described by the equation \(x + 5y - z = 25\). The line of intersection of these two planes can be found by solving these simultaneous equations. Once we have the line, we can determine a vector that is parallel to it. This concept is common in linear algebra and vector calculus, where understanding intersections and vector directions can be applied in fields such as physics, engineering, and computer graphics.
Transcribed Image Text:**Question 28:** Find a vector parallel to the line of intersection of the two planes \(4x - 3y + 2z = 12\) and \(x + 5y - z = 25\). --- **Explanation for Educational Context:** In this question, we are tasked with finding a vector that runs parallel to the line where two planes intersect. Each plane is described by its own linear equation in three-dimensional space: - The first plane is represented by the equation \(4x - 3y + 2z = 12\). - The second plane is described by the equation \(x + 5y - z = 25\). The line of intersection of these two planes can be found by solving these simultaneous equations. Once we have the line, we can determine a vector that is parallel to it. This concept is common in linear algebra and vector calculus, where understanding intersections and vector directions can be applied in fields such as physics, engineering, and computer graphics.
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