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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![### Find the Distance Between Two Lines
Given the following vector equations of lines:
\[
\vec{r}_1 = \langle 1, 1, 0 \rangle + t \langle 1, 6, 2 \rangle
\]
\[
\vec{r}_2 = \langle 1, 5, -2 \rangle + s \langle 2, 15, 6 \rangle
\]
We are tasked with finding the distance between these two lines.
### Explanation:
- \(\vec{r}_1\) is the vector equation of the first line.
- Here, \(\langle 1, 1, 0 \rangle\) is a point on the first line.
- \(\langle 1, 6, 2 \rangle\) is the direction vector of the first line.
- \(t\) is a scalar parameter which, when varied, traces out the points along the line.
- \(\vec{r}_2\) is the vector equation of the second line.
- Here, \(\langle 1, 5, -2 \rangle\) is a point on the second line.
- \(\langle 2, 15, 6 \rangle\) is the direction vector of the second line.
- \(s\) is a scalar parameter which, when varied, traces out the points along the line.
To find the distance between these two lines, it involves using the formula for the distance between skew lines.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F416064f9-c57b-46ee-9b50-14b257515654%2F44a5bd3b-55da-4955-971e-24c9fdedcba8%2Fi0ulqn_processed.png&w=3840&q=75)
Transcribed Image Text:### Find the Distance Between Two Lines
Given the following vector equations of lines:
\[
\vec{r}_1 = \langle 1, 1, 0 \rangle + t \langle 1, 6, 2 \rangle
\]
\[
\vec{r}_2 = \langle 1, 5, -2 \rangle + s \langle 2, 15, 6 \rangle
\]
We are tasked with finding the distance between these two lines.
### Explanation:
- \(\vec{r}_1\) is the vector equation of the first line.
- Here, \(\langle 1, 1, 0 \rangle\) is a point on the first line.
- \(\langle 1, 6, 2 \rangle\) is the direction vector of the first line.
- \(t\) is a scalar parameter which, when varied, traces out the points along the line.
- \(\vec{r}_2\) is the vector equation of the second line.
- Here, \(\langle 1, 5, -2 \rangle\) is a point on the second line.
- \(\langle 2, 15, 6 \rangle\) is the direction vector of the second line.
- \(s\) is a scalar parameter which, when varied, traces out the points along the line.
To find the distance between these two lines, it involves using the formula for the distance between skew lines.
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