Find the equation for the plane that passes through (2, 4, -1) and contains the intersection of the planes $ x - y - 4z = 2 $ and $ -2x + y + 2z = 3 $. (Hint: Find the parametric equation of the intersection of the two lines, choose two points on that line and then use the given point to write the equation of the plane passing through three points).

Answered: Find the equation for the plane that…

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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Question

Find the equation for the plane that passes through (2, 4, -1) and contains the intersection of the planes $ x - y - 4z = 2 $ and $ -2x + y + 2z = 3 $. (Hint: Find the parametric equation of the intersection of the two lines, choose two points on that line and then use the given point to write the equation of the plane passing through three points).

Expert Solution

Step 1

The plane equation $A x + B y + C z + D = 0$ passing through three points $(x_{1}, y_{1}, z_{1})$ , $(x_{2}, y_{2}, z_{2})$ and $(x_{3}, y_{3}, z_{3})$ is obtained by the following formula.

$\begin{array}{rcl} A & = & |\begin{matrix} y_{1} & z_{1} & 1 \\ y_{2} & z_{2} & 1 \\ y_{3} & z_{3} & 1 \end{matrix}| \\ B & = & - |\begin{matrix} x_{1} & z_{1} & 1 \\ x_{2} & z_{2} & 1 \\ x_{3} & z_{3} & 1 \end{matrix}| \\ C & = & |\begin{matrix} x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \end{matrix}| \\ D & = & - |\begin{matrix} x_{1} & y_{1} & z_{1} \\ x_{2} & y_{2} & z_{2} \\ x_{3} & y_{3} & z_{3} \end{matrix}| \end{array}$