Find parametric equations for the line. (Use the parameter t.) The line through (-8, 4, 5) and parallel to the line x =y = z + 1 2 (x(t), y(t), z(t)) = Find the symmetric equations. x + 8 - Y- 4 = z - 5 2 3 x + 8 y + 4 = z + 5 3 Y = z - 5 2 X = L = z + 5 %3D 2 3 X - 8 y + 4 = z + 5 3. 2.
Find parametric equations for the line. (Use the parameter t.) The line through (-8, 4, 5) and parallel to the line x =y = z + 1 2 (x(t), y(t), z(t)) = Find the symmetric equations. x + 8 - Y- 4 = z - 5 2 3 x + 8 y + 4 = z + 5 3 Y = z - 5 2 X = L = z + 5 %3D 2 3 X - 8 y + 4 = z + 5 3. 2.
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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![### Parametric and Symmetric Equations of a Line
#### Problem Statement:
**Find parametric equations for the line. (Use the parameter t.)**
Given the line passing through the point (-8, 4, 5) and parallel to the line defined by the equations:
\[ \frac{1}{2}x = \frac{1}{3}y = z + 1 \]
We need to find the parametric form \((x(t), y(t), z(t))\).
#### Steps to Find the Parametric Equations:
1. **Identify the direction ratios of the line given in symmetric form:**
- The given symmetric form of the line is \(\frac{x}{2} = \frac{y}{3} = z + 1\).
- This shows that direction ratios (proportionality constants) are \(2\) for \(x\), \(3\) for \(y\), and \(1\) for \(z\).
2. **Develop the parametric equations using the point (-8, 4, 5):**
- Using these direction ratios, the equations are:
\[
x(t) = -8 + 2t \\
y(t) = 4 + 3t \\
z(t) = 5 + t
\]
3. **Write the parametric equations explicitly:**
\[
\left( x(t), y(t), z(t) \right) = \left( -8 + 2t, 4 + 3t, 5 + t \right)
\]
#### Select the Symmetric Equations:
**Find the symmetric equations for the line:**
Given the parametric equations, we convert them to symmetric form:
\[
t = \frac{x + 8}{2} = \frac{y - 4}{3} = z - 5
\]
**Options for symmetric form:**
- \(\frac{x + 8}{2} = \frac{y - 4}{3} = z - 5\) (correct)
- \(\frac{x + 8}{2} = \frac{y + 4}{3} = z + 5\) (incorrect)
- \(\frac{x}{2} = \frac{y}{3} = z - 5\) (incorrect)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2ba0aec5-65e2-43b8-bb6a-70de8eae6b73%2F58ec53a0-b31d-4458-bacf-499926d16d81%2Frovf3k3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Parametric and Symmetric Equations of a Line
#### Problem Statement:
**Find parametric equations for the line. (Use the parameter t.)**
Given the line passing through the point (-8, 4, 5) and parallel to the line defined by the equations:
\[ \frac{1}{2}x = \frac{1}{3}y = z + 1 \]
We need to find the parametric form \((x(t), y(t), z(t))\).
#### Steps to Find the Parametric Equations:
1. **Identify the direction ratios of the line given in symmetric form:**
- The given symmetric form of the line is \(\frac{x}{2} = \frac{y}{3} = z + 1\).
- This shows that direction ratios (proportionality constants) are \(2\) for \(x\), \(3\) for \(y\), and \(1\) for \(z\).
2. **Develop the parametric equations using the point (-8, 4, 5):**
- Using these direction ratios, the equations are:
\[
x(t) = -8 + 2t \\
y(t) = 4 + 3t \\
z(t) = 5 + t
\]
3. **Write the parametric equations explicitly:**
\[
\left( x(t), y(t), z(t) \right) = \left( -8 + 2t, 4 + 3t, 5 + t \right)
\]
#### Select the Symmetric Equations:
**Find the symmetric equations for the line:**
Given the parametric equations, we convert them to symmetric form:
\[
t = \frac{x + 8}{2} = \frac{y - 4}{3} = z - 5
\]
**Options for symmetric form:**
- \(\frac{x + 8}{2} = \frac{y - 4}{3} = z - 5\) (correct)
- \(\frac{x + 8}{2} = \frac{y + 4}{3} = z + 5\) (incorrect)
- \(\frac{x}{2} = \frac{y}{3} = z - 5\) (incorrect)
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