Determine if the lines intersect, are parallel, are skew (don't intersect but aren't parallel) or are the same: L₁: x = 6 + 2t L₂: x = 4 - 8t y = 3 - 4t y = 1 + 16t z = 5 + t z = 6 - 4t O intersect O parallel skew O same line
Determine if the lines intersect, are parallel, are skew (don't intersect but aren't parallel) or are the same: L₁: x = 6 + 2t L₂: x = 4 - 8t y = 3 - 4t y = 1 + 16t z = 5 + t z = 6 - 4t O intersect O parallel skew O same line
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Line Intersection and Parallelism
#### Question:
Determine if the lines intersect, are parallel, are skew (don’t intersect but aren’t parallel), or are the same:
\[ L_1: \begin{cases} x = -6 + 2t \\ y = 3 - 4t \\ z = 5 + t \end{cases} \]
\[ L_2: \begin{cases} x = -4 - 8t \\ y = -1 + 16t \\ z = 6 - 4t \end{cases} \]
#### Options:
- ⃝ intersect
- ⃝ parallel
- ⃝ skew
- ⃝ same line
---
This mathematical exercise is designed to test your understanding of how to determine the relationships between two lines in 3-dimensional space. Specifically, you need to identify whether the given parametric equations for two lines suggest that the lines intersect at a point, run parallel to each other, are skew (do not intersect while also not being parallel), or are actually the same line.
To solve this problem, you will typically compare the direction vectors of the lines and then check for any possible points of intersection or confirmation of identical line equations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F55815b3c-2575-4201-a9dc-5a4b41a14c1d%2F7dc55292-301c-48c3-b743-051b4fddeaa9%2Ffiwi5kb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
### Line Intersection and Parallelism
#### Question:
Determine if the lines intersect, are parallel, are skew (don’t intersect but aren’t parallel), or are the same:
\[ L_1: \begin{cases} x = -6 + 2t \\ y = 3 - 4t \\ z = 5 + t \end{cases} \]
\[ L_2: \begin{cases} x = -4 - 8t \\ y = -1 + 16t \\ z = 6 - 4t \end{cases} \]
#### Options:
- ⃝ intersect
- ⃝ parallel
- ⃝ skew
- ⃝ same line
---
This mathematical exercise is designed to test your understanding of how to determine the relationships between two lines in 3-dimensional space. Specifically, you need to identify whether the given parametric equations for two lines suggest that the lines intersect at a point, run parallel to each other, are skew (do not intersect while also not being parallel), or are actually the same line.
To solve this problem, you will typically compare the direction vectors of the lines and then check for any possible points of intersection or confirmation of identical line equations.
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