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
![**Problem:**
Verify that the points are the vertices of a parallelogram, and find its area.
Points:
- \( A(1, 1, 3) \)
- \( B(10, -7, 8) \)
- \( C(12, -4, 1) \)
- \( D(3, 4, -4) \)
**Solution:**
To determine if these points form the vertices of a parallelogram, you should check if opposite sides are equal:
1. Calculate the distance between points to verify opposite sides are equal:
- \( AB \) and \( CD \)
- \( BC \) and \( DA \)
2. Use the distance formula for three-dimensional space:
\[
\text{Distance} = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}
\]
3. Compute the vectors for sides and confirm their equality to establish the parallelogram.
4. After confirming a parallelogram, use the vector cross product and the area formula to find the area:
\[
\text{Area} = \| \mathbf{v} \times \mathbf{w} \|
\]
where \(\mathbf{v}\) and \(\mathbf{w}\) are vectors representing adjacent sides.
5. Calculate the magnitude of the cross product to get the area.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba13667f-f60c-45eb-87f2-7662478f2e6d%2Fb843fbf5-44ff-462f-84d8-863e90157550%2Fwcqnb27_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem:**
Verify that the points are the vertices of a parallelogram, and find its area.
Points:
- \( A(1, 1, 3) \)
- \( B(10, -7, 8) \)
- \( C(12, -4, 1) \)
- \( D(3, 4, -4) \)
**Solution:**
To determine if these points form the vertices of a parallelogram, you should check if opposite sides are equal:
1. Calculate the distance between points to verify opposite sides are equal:
- \( AB \) and \( CD \)
- \( BC \) and \( DA \)
2. Use the distance formula for three-dimensional space:
\[
\text{Distance} = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}
\]
3. Compute the vectors for sides and confirm their equality to establish the parallelogram.
4. After confirming a parallelogram, use the vector cross product and the area formula to find the area:
\[
\text{Area} = \| \mathbf{v} \times \mathbf{w} \|
\]
where \(\mathbf{v}\) and \(\mathbf{w}\) are vectors representing adjacent sides.
5. Calculate the magnitude of the cross product to get the area.
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