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3) The line L: x = 3 + 2t, y = 2t, z = t_intersects the plane x + 3y -z = −4 in a point P. Find the coordinates of P and find equations for the line in the plane through P perpendicular to L.

3) The line L: x = 3 + 2t, y = 2t, z = t_intersects the plane x + 3y -z = −4 in a point P. Find the coordinates of P and find equations for the line in the plane through P perpendicular to L.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Problem Statement Given: - The line \( L: x = 3 + 2t, \; y = 2t, \; z = t \) This line intersects the plane \( x + 3y - z = -4 \) at a point \( P \). Tasks: 1. Find the coordinates of point \( P \), where the line intersects the plane. 2. Determine the equations for the line in the plane through point \( P \) that is perpendicular to line \( L \). **Solution Approach:** 1. **Finding the coordinates of \( P \):** - Substitute the parametric equations of the line into the plane equation: \( (3 + 2t) + 3(2t) - t = -4 \). - Solve for \( t \) and then find \( x, y, z \) using the values of \( t \). 2. **Equation for the line in the plane through \( P \) perpendicular to \( L \):** - Determine the direction vector of line \( L \). - Use the direction vector of \( L \) and normal to the plane to find the direction vector of the new line. - Formulate the parametric equations for the new line using point \( P \). **Detailed Explanations & Calculations:** This section should include step-by-step calculations, showing the process of algebraic manipulations to find the coordinates of \( P \) and the parametric equations of the perpendicular line.
Transcribed Image Text:### Problem Statement Given: - The line \( L: x = 3 + 2t, \; y = 2t, \; z = t \) This line intersects the plane \( x + 3y - z = -4 \) at a point \( P \). Tasks: 1. Find the coordinates of point \( P \), where the line intersects the plane. 2. Determine the equations for the line in the plane through point \( P \) that is perpendicular to line \( L \). **Solution Approach:** 1. **Finding the coordinates of \( P \):** - Substitute the parametric equations of the line into the plane equation: \( (3 + 2t) + 3(2t) - t = -4 \). - Solve for \( t \) and then find \( x, y, z \) using the values of \( t \). 2. **Equation for the line in the plane through \( P \) perpendicular to \( L \):** - Determine the direction vector of line \( L \). - Use the direction vector of \( L \) and normal to the plane to find the direction vector of the new line. - Formulate the parametric equations for the new line using point \( P \). **Detailed Explanations & Calculations:** This section should include step-by-step calculations, showing the process of algebraic manipulations to find the coordinates of \( P \) and the parametric equations of the perpendicular line.
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