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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:**
1. Find an equation for the plane passing through the point (3, 1, 2) with normal vector \(i + 2j + 7k\). Put the answer into the form \(Ax + By + Cz = D\).
**Solution Approach:**
To find the equation of a plane given a point and a normal vector, you can use the following general formula for the equation of a plane:
\[ A(x - x_0) + B(y - y_0) + C(z - z_0) = 0 \]
where \((x_0, y_0, z_0)\) is the given point and \(A\), \(B\), and \(C\) are the components of the normal vector.
Given:
- Point: \((x_0, y_0, z_0) = (3, 1, 2)\)
- Normal vector: \(\langle A, B, C \rangle = \langle 1, 2, 7 \rangle\)
Plug these values into the formula to obtain the equation of the plane.
**Detailed Explanation:**
Substitute the values into the plane equation:
\[ 1(x - 3) + 2(y - 1) + 7(z - 2) = 0 \]
Simplify the equation:
\[ (x - 3) + 2(y - 1) + 7(z - 2) = 0 \]
\[ x - 3 + 2y - 2 + 7z - 14 = 0 \]
\[ x + 2y + 7z - 19 = 0 \]
Thus, the equation of the plane is:
\[ x + 2y + 7z = 19 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc740e8b2-d410-40ba-b674-1beb1e9bb454%2Fcdee86c8-f295-48df-91d8-12060735b342%2Fszrd5ztm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
1. Find an equation for the plane passing through the point (3, 1, 2) with normal vector \(i + 2j + 7k\). Put the answer into the form \(Ax + By + Cz = D\).
**Solution Approach:**
To find the equation of a plane given a point and a normal vector, you can use the following general formula for the equation of a plane:
\[ A(x - x_0) + B(y - y_0) + C(z - z_0) = 0 \]
where \((x_0, y_0, z_0)\) is the given point and \(A\), \(B\), and \(C\) are the components of the normal vector.
Given:
- Point: \((x_0, y_0, z_0) = (3, 1, 2)\)
- Normal vector: \(\langle A, B, C \rangle = \langle 1, 2, 7 \rangle\)
Plug these values into the formula to obtain the equation of the plane.
**Detailed Explanation:**
Substitute the values into the plane equation:
\[ 1(x - 3) + 2(y - 1) + 7(z - 2) = 0 \]
Simplify the equation:
\[ (x - 3) + 2(y - 1) + 7(z - 2) = 0 \]
\[ x - 3 + 2y - 2 + 7z - 14 = 0 \]
\[ x + 2y + 7z - 19 = 0 \]
Thus, the equation of the plane is:
\[ x + 2y + 7z = 19 \]
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