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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13.1.21 With the attached image, please help me answer the questions.
![**Title: Calculating the Area of a Triangle in 3D Space**
**Objective:**
Determine the area of triangle \( T \) with given vertex coordinates using vector cross product.
**Problem Statement:**
Find the area of the following triangle \( T \).
**Vertex Coordinates:**
- \( A(6, 6, 3) \)
- \( B(7, 16, 5) \)
- \( C(7, 7, 3) \)
**Instructions:**
1. **Vectors Definition:**
- Define side \( AB \) as vector \( \mathbf{u} \).
- Define side \( AC \) as vector \( \mathbf{v} \).
2. **Cross Product Calculation:**
- Calculate the cross product of vectors \( \mathbf{u} \) and \( \mathbf{v} \) as follows:
\[
\mathbf{u} \times \mathbf{v} = (\Box) \mathbf{i} + (\Box) \mathbf{j} + (\Box) \mathbf{k}
\]
- Simplify your answers.
**Note:**
The magnitude of the cross product gives twice the area of the triangle. Use this property to find the area of the triangle \( T \).
---
This problem involves using 3D vector algebra to find the area of a triangle given its vertices in space. By calculating the cross product of two sides and finding its magnitude, one can determine the area efficiently.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fabd5e741-0235-409f-80de-883f5f0e5da7%2F83479014-9595-403a-89ca-892414d4db7a%2F0r8kfz_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Calculating the Area of a Triangle in 3D Space**
**Objective:**
Determine the area of triangle \( T \) with given vertex coordinates using vector cross product.
**Problem Statement:**
Find the area of the following triangle \( T \).
**Vertex Coordinates:**
- \( A(6, 6, 3) \)
- \( B(7, 16, 5) \)
- \( C(7, 7, 3) \)
**Instructions:**
1. **Vectors Definition:**
- Define side \( AB \) as vector \( \mathbf{u} \).
- Define side \( AC \) as vector \( \mathbf{v} \).
2. **Cross Product Calculation:**
- Calculate the cross product of vectors \( \mathbf{u} \) and \( \mathbf{v} \) as follows:
\[
\mathbf{u} \times \mathbf{v} = (\Box) \mathbf{i} + (\Box) \mathbf{j} + (\Box) \mathbf{k}
\]
- Simplify your answers.
**Note:**
The magnitude of the cross product gives twice the area of the triangle. Use this property to find the area of the triangle \( T \).
---
This problem involves using 3D vector algebra to find the area of a triangle given its vertices in space. By calculating the cross product of two sides and finding its magnitude, one can determine the area efficiently.
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