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3. The plane in ' with equation 4x=34 %3D

3. The plane in ' with equation 4x=34 %3D

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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Find the basis of the following subspace.
**Transcription for Educational Website:** --- **The Plane in \(\mathbb{R}^3\):** Consider a plane in three-dimensional space \(\mathbb{R}^3\) defined by the equation: \[ z = 4x - 3y \] **Explanation:** This equation represents a plane in \(\mathbb{R}^3\), where \(z\) is expressed in terms of \(x\) and \(y\). The coefficients determine the orientation and slope of the plane. Specifically, this plane slopes upward in the direction of the positive \(x\)-axis with a gradient of 4, and downward with a gradient of -3 in the direction of the positive \(y\)-axis. (Note: There are no graphs or diagrams included in the text to be explained further.) ---
Transcribed Image Text:**Transcription for Educational Website:** --- **The Plane in \(\mathbb{R}^3\):** Consider a plane in three-dimensional space \(\mathbb{R}^3\) defined by the equation: \[ z = 4x - 3y \] **Explanation:** This equation represents a plane in \(\mathbb{R}^3\), where \(z\) is expressed in terms of \(x\) and \(y\). The coefficients determine the orientation and slope of the plane. Specifically, this plane slopes upward in the direction of the positive \(x\)-axis with a gradient of 4, and downward with a gradient of -3 in the direction of the positive \(y\)-axis. (Note: There are no graphs or diagrams included in the text to be explained further.) ---
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