Find parametric equations for the line. (Use the parameter t.) The line through the points (0,, 1) and (9, 1, –4) (x(t), y(t), z(t)) =| )- Find the symmetric equations. X - 9 - 2y - 2 = z + 4 -5 – 9. -5 z + 4 2x - 2 = Y-2 = x + 4 = 2y – 2 = Z - 9 -5 9 + 9x = 1 + = -4 - 5z x - 9 = 2y - 2 = z + 4
Find parametric equations for the line. (Use the parameter t.) The line through the points (0,, 1) and (9, 1, –4) (x(t), y(t), z(t)) =| )- Find the symmetric equations. X - 9 - 2y - 2 = z + 4 -5 – 9. -5 z + 4 2x - 2 = Y-2 = x + 4 = 2y – 2 = Z - 9 -5 9 + 9x = 1 + = -4 - 5z x - 9 = 2y - 2 = z + 4
Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter9: Vectors In Two And Three Dimensions
Section9.6: Equations Of Lines And Planes
Problem 37E
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![**Finding Parametric and Symmetric Equations for a Line**
**Problem Statement:**
Given two points on a line, find the parametric equations using the parameter \( t \) and the symmetric equations.
Points provided:
1. \( (0, \frac{1}{2}, 1) \)
2. \( (9, 1, -4) \)
### Parametric Equations
To find the parametric equations for the line through these points, we use the parameter \( t \). The general form for the parametric equations is:
\[ (x(t), y(t), z(t)) = (x_0 + at, y_0 + bt, z_0 + ct) \]
where \( (x_0, y_0, z_0) \) is the initial point and \( (a, b, c) \) represents the direction ratios found by subtracting the coordinates of the two points.
### Symmetric Equations
To find the symmetric equations of the line, we use the direction ratios obtained previously. The symmetric form of the equation for the line is:
\[\frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c}\]
### Step-by-Step Solution:
1. **Parameter Form Calculation:**
- **Initial Point (x_0, y_0, z_0):** \((0, \frac{1}{2}, 1)\)
- **Direction Ratios (a, b, c):**
\[
a = 9 - 0 = 9 \\
b = 1 - \frac{1}{2} = \frac{1}{2} \\
c = -4 - 1 = -5
\]
Therefore, the parametric equations combine to give:
\[
(x(t), y(t), z(t)) = \left(0 + 9t, \frac{1}{2} + \frac{1}{2}t, 1 - 5t \right)
\]
2. **Symmetric Form Calculation:**
Using the direction ratios:
\[
\frac{x - 0}{9} = \frac{y - \frac{1}{2}}{\frac{1}{2}} = \frac{z - 1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2ba0aec5-65e2-43b8-bb6a-70de8eae6b73%2F6030eb45-dfe9-46d1-a1f6-05a7cea69f2f%2F6s7cwr5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding Parametric and Symmetric Equations for a Line**
**Problem Statement:**
Given two points on a line, find the parametric equations using the parameter \( t \) and the symmetric equations.
Points provided:
1. \( (0, \frac{1}{2}, 1) \)
2. \( (9, 1, -4) \)
### Parametric Equations
To find the parametric equations for the line through these points, we use the parameter \( t \). The general form for the parametric equations is:
\[ (x(t), y(t), z(t)) = (x_0 + at, y_0 + bt, z_0 + ct) \]
where \( (x_0, y_0, z_0) \) is the initial point and \( (a, b, c) \) represents the direction ratios found by subtracting the coordinates of the two points.
### Symmetric Equations
To find the symmetric equations of the line, we use the direction ratios obtained previously. The symmetric form of the equation for the line is:
\[\frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c}\]
### Step-by-Step Solution:
1. **Parameter Form Calculation:**
- **Initial Point (x_0, y_0, z_0):** \((0, \frac{1}{2}, 1)\)
- **Direction Ratios (a, b, c):**
\[
a = 9 - 0 = 9 \\
b = 1 - \frac{1}{2} = \frac{1}{2} \\
c = -4 - 1 = -5
\]
Therefore, the parametric equations combine to give:
\[
(x(t), y(t), z(t)) = \left(0 + 9t, \frac{1}{2} + \frac{1}{2}t, 1 - 5t \right)
\]
2. **Symmetric Form Calculation:**
Using the direction ratios:
\[
\frac{x - 0}{9} = \frac{y - \frac{1}{2}}{\frac{1}{2}} = \frac{z - 1
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