Let line L: x = 2t, y = 0, z = 3, and let line M: x = 0, y = 8 + s, z = 7+s, for some constants t and s. Determine whether the lines are equal, parallel but not equal, intersect, or are skew.
Let line L: x = 2t, y = 0, z = 3, and let line M: x = 0, y = 8 + s, z = 7+s, for some constants t and s. Determine whether the lines are equal, parallel but not equal, intersect, or are skew.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.3: Lines
Problem 33E
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![**Problem Statement:**
Let line \( L: x = 2t, y = 0, z = 3 \), and let line \( M: x = 0, y = 8 + s, z = 7 + s \), for some constants \( t \) and \( s \). Determine whether the lines are equal, parallel but not equal, intersect, or are skew.
---
**Explanation:**
1. **Line \( L \) Representation:**
- Parametric Equations:
\[
x = 2t, \quad y = 0, \quad z = 3
\]
- This describes a line where \( y \) is always 0 and \( z \) is always 3. The variation in \( x \) is controlled by the parameter \( t \).
2. **Line \( M \) Representation:**
- Parametric Equations:
\[
x = 0, \quad y = 8 + s, \quad z = 7 + s
\]
- This describes a line where \( x \) is always 0. The variation in \( y \) and \( z \) is controlled by the parameter \( s \).
3. **Comparison between Line \( L \) and Line \( M \):**
- Check if lines are the same (equal):
- Lines \( L \) and \( M \) have different parametric forms; they are not identical.
- Check if lines are parallel:
- To be parallel, the direction ratios should be proportional, but here, the parametric form of line \( L \) (given by \(\langle 2, 0, 0 \rangle\)) is different from the direction ratios of line \( M \) (given by \(\langle 0, 1, 1 \rangle\)). Hence, the lines are not parallel.
- Check if lines intersect:
- For lines to intersect, there should be a common point that satisfies both lines' equations.
- Solving \(2t = 0\) and checking common \( y \) and \( z \) values will show no common \( x \), \( y \), and \( z \) values simultaneously satisfy both lines' equations.
- Check if lines are skew:
- Since lines are not parallel and](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F58097f1d-9f7f-481d-be94-f210527e0168%2F2b4003a0-b69e-427e-9f54-4b5149081c8e%2Fc08kkpc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Let line \( L: x = 2t, y = 0, z = 3 \), and let line \( M: x = 0, y = 8 + s, z = 7 + s \), for some constants \( t \) and \( s \). Determine whether the lines are equal, parallel but not equal, intersect, or are skew.
---
**Explanation:**
1. **Line \( L \) Representation:**
- Parametric Equations:
\[
x = 2t, \quad y = 0, \quad z = 3
\]
- This describes a line where \( y \) is always 0 and \( z \) is always 3. The variation in \( x \) is controlled by the parameter \( t \).
2. **Line \( M \) Representation:**
- Parametric Equations:
\[
x = 0, \quad y = 8 + s, \quad z = 7 + s
\]
- This describes a line where \( x \) is always 0. The variation in \( y \) and \( z \) is controlled by the parameter \( s \).
3. **Comparison between Line \( L \) and Line \( M \):**
- Check if lines are the same (equal):
- Lines \( L \) and \( M \) have different parametric forms; they are not identical.
- Check if lines are parallel:
- To be parallel, the direction ratios should be proportional, but here, the parametric form of line \( L \) (given by \(\langle 2, 0, 0 \rangle\)) is different from the direction ratios of line \( M \) (given by \(\langle 0, 1, 1 \rangle\)). Hence, the lines are not parallel.
- Check if lines intersect:
- For lines to intersect, there should be a common point that satisfies both lines' equations.
- Solving \(2t = 0\) and checking common \( y \) and \( z \) values will show no common \( x \), \( y \), and \( z \) values simultaneously satisfy both lines' equations.
- Check if lines are skew:
- Since lines are not parallel and
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