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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Question
3 on paper please
![**Problem 3: Finding the Equation of a Plane Using Vector Operations**
**Given:**
Find an equation of the plane passing through the points \((1, 2, -6)\), \((2, 3, -2)\), and \((-1, 4, 8)\). Use vector operations.
**Solution Outline:**
To find the equation of a plane passing through three points \((x_1, y_1, z_1)\), \((x_2, y_2, z_2)\), and \((x_3, y_3, z_3)\):
1. Find two vectors that lie in the plane.
2. Compute the cross product of these vectors to find a normal vector to the plane.
3. Use the normal vector and one of the points to write the equation of the plane in the form \(Ax + By + Cz = D\).
**Steps:**
1. **Identify the points:**
Points: \(A = (1, 2, -6)\), \(B = (2, 3, -2)\), \(C = (-1, 4, 8)\).
2. **Find two vectors that lie in the plane:**
Let \(\vec{AB}\) and \(\vec{AC}\) be the two vectors.
\[
\vec{AB} = B - A = (2 - 1, 3 - 2, -2 + 6) = (1, 1, 4)
\]
\[
\vec{AC} = C - A = (-1 - 1, 4 - 2, 8 + 6) = (-2, 2, 14)
\]
3. **Compute the cross product of \(\vec{AB}\) and \(\vec{AC}\) to find the normal vector \(\vec{n}\):**
\[
\vec{n} = \vec{AB} \times \vec{AC} =
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
1 & 1 & 4 \\
-2 & 2 & 14
\end{vmatrix}
\]
This gives us:
\[
\vec{n} = (1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F48922dd6-b6f8-4fac-84c6-05a7db5750f6%2F76d7e58c-107e-4669-8a2f-7409cdea342a%2Fxxbmbnl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 3: Finding the Equation of a Plane Using Vector Operations**
**Given:**
Find an equation of the plane passing through the points \((1, 2, -6)\), \((2, 3, -2)\), and \((-1, 4, 8)\). Use vector operations.
**Solution Outline:**
To find the equation of a plane passing through three points \((x_1, y_1, z_1)\), \((x_2, y_2, z_2)\), and \((x_3, y_3, z_3)\):
1. Find two vectors that lie in the plane.
2. Compute the cross product of these vectors to find a normal vector to the plane.
3. Use the normal vector and one of the points to write the equation of the plane in the form \(Ax + By + Cz = D\).
**Steps:**
1. **Identify the points:**
Points: \(A = (1, 2, -6)\), \(B = (2, 3, -2)\), \(C = (-1, 4, 8)\).
2. **Find two vectors that lie in the plane:**
Let \(\vec{AB}\) and \(\vec{AC}\) be the two vectors.
\[
\vec{AB} = B - A = (2 - 1, 3 - 2, -2 + 6) = (1, 1, 4)
\]
\[
\vec{AC} = C - A = (-1 - 1, 4 - 2, 8 + 6) = (-2, 2, 14)
\]
3. **Compute the cross product of \(\vec{AB}\) and \(\vec{AC}\) to find the normal vector \(\vec{n}\):**
\[
\vec{n} = \vec{AB} \times \vec{AC} =
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
1 & 1 & 4 \\
-2 & 2 & 14
\end{vmatrix}
\]
This gives us:
\[
\vec{n} = (1
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