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.

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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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
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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