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
#4
![**Problem 4:**
Write the **parametric** and **symmetric** equation of the line passing through the point \((-3, 5, 4)\) in the direction of \(\langle 3, -2, 1 \rangle\).
**Solution:**
1. **Parametric Equations:**
Given the point \((-3, 5, 4)\) and direction vector \(\langle 3, -2, 1 \rangle\), the parametric equations for the line are:
\[
x = -3 + 3t
\]
\[
y = 5 - 2t
\]
\[
z = 4 + t
\]
where \(t\) is the parameter.
2. **Symmetric Equation:**
The symmetric form of the line can be derived from the parametric equations as follows:
\[
\frac{x + 3}{3} = \frac{y - 5}{-2} = \frac{z - 4}{1}
\]
These equations provide the parametric and symmetric forms of the line in three-dimensional space.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdc238911-2b79-470a-ae35-fd09314a6a93%2F115ccee1-ddef-4bed-a4b1-5ebefa75ad2c%2Fhy2cdik_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 4:**
Write the **parametric** and **symmetric** equation of the line passing through the point \((-3, 5, 4)\) in the direction of \(\langle 3, -2, 1 \rangle\).
**Solution:**
1. **Parametric Equations:**
Given the point \((-3, 5, 4)\) and direction vector \(\langle 3, -2, 1 \rangle\), the parametric equations for the line are:
\[
x = -3 + 3t
\]
\[
y = 5 - 2t
\]
\[
z = 4 + t
\]
where \(t\) is the parameter.
2. **Symmetric Equation:**
The symmetric form of the line can be derived from the parametric equations as follows:
\[
\frac{x + 3}{3} = \frac{y - 5}{-2} = \frac{z - 4}{1}
\]
These equations provide the parametric and symmetric forms of the line in three-dimensional space.
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