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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![**Problem Statement:**
Find the equation of the plane through the points (1, -1, 0), (1, 0, 1), and (2, 2, 1).
---
To solve this problem, you need to find the equation of the plane that passes through three given points in 3-dimensional space. The general equation for a plane in 3D is given by:
\[ ax + by + cz = d \]
Where \(a\), \(b\), and \(c\) are the coefficients that define the orientation of the plane, and \(d\) is a constant term.
Given points:
1. \( A(1, -1, 0) \)
2. \( B(1, 0, 1) \)
3. \( C(2, 2, 1) \)
Steps to determine the plane equation:
1. Find two vectors that lie on the plane using the given points.
2. Calculate the normal vector to the plane by finding the cross product of the obtained vectors.
3. Use the normal vector and one of the points to determine the equation of the plane.
1. Form two vectors \( \vec{AB} \) and \( \vec{AC} \):
\[
\vec{AB} = B - A = (1 - 1, 0 - (-1), 1 - 0) = (0, 1, 1)
\]
\[
\vec{AC} = C - A = (2 - 1, 2 - (-1), 1 - 0) = (1, 3, 1)
\]
2. Calculate the normal vector \( \vec{n} \) by finding the cross product of \( \vec{AB} \) and \( \vec{AC} \):
\[
\vec{n} = \vec{AB} \times \vec{AC} =
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
0 & 1 & 1 \\
1 & 3 & 1
\end{vmatrix}
\]
The determinant gives us:
\[
\vec{n} = \mathbf{i}(1*1 - 1*3) - \mathbf{j}(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0bda9c1d-08ea-450e-a130-838d16131279%2Fcd18760f-a16a-473d-90e1-bf7e8942ad81%2Fonods88_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the equation of the plane through the points (1, -1, 0), (1, 0, 1), and (2, 2, 1).
---
To solve this problem, you need to find the equation of the plane that passes through three given points in 3-dimensional space. The general equation for a plane in 3D is given by:
\[ ax + by + cz = d \]
Where \(a\), \(b\), and \(c\) are the coefficients that define the orientation of the plane, and \(d\) is a constant term.
Given points:
1. \( A(1, -1, 0) \)
2. \( B(1, 0, 1) \)
3. \( C(2, 2, 1) \)
Steps to determine the plane equation:
1. Find two vectors that lie on the plane using the given points.
2. Calculate the normal vector to the plane by finding the cross product of the obtained vectors.
3. Use the normal vector and one of the points to determine the equation of the plane.
1. Form two vectors \( \vec{AB} \) and \( \vec{AC} \):
\[
\vec{AB} = B - A = (1 - 1, 0 - (-1), 1 - 0) = (0, 1, 1)
\]
\[
\vec{AC} = C - A = (2 - 1, 2 - (-1), 1 - 0) = (1, 3, 1)
\]
2. Calculate the normal vector \( \vec{n} \) by finding the cross product of \( \vec{AB} \) and \( \vec{AC} \):
\[
\vec{n} = \vec{AB} \times \vec{AC} =
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
0 & 1 & 1 \\
1 & 3 & 1
\end{vmatrix}
\]
The determinant gives us:
\[
\vec{n} = \mathbf{i}(1*1 - 1*3) - \mathbf{j}(
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