Question

Find parametric equations and symmetric equations for the line. The line through the points $(12,9,-13)$ and $(-7,9,11)$

Name: Problem 9, Chapter 12: Calculus: Early Transcendentals, Metric Edition
Uploaded: 2022-09-26T23:29:03-08:00
Duration: 3 min 51 s
Channel: Steven Clarke
Description: Find parametric equations and symmetric equations for the line. The line through the points (12,9,-13) and (-7,9,11)

   Find parametric equations and symmetric equations for the line.
The line through the points $(12,9,-13)$ and $(-7,9,11)$

Calculus: Early Transcendentals, Metric Edition

James Stewart,… 9th Edition

Chapter 12, Problem 9

Instant Answer

Solved by Expert Steven Clarke

09/26/2022

Step 1

This can be found by subtracting the coordinates of the two given points. Let's denote the points as $A(12,9,-13)$ and $B(-7,9,11)$. The direction vector $\vec{AB}$ is given by $\vec{OB} - \vec{OA}$, where $\vec{OB}$ and $\vec{OA}$ are position vectors of points B Show more…

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Find parametric equations and symmetric equations for the line. The line through the points $(12,9,-13)$ and $(-7,9,11)$

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Key Concepts

Direction Vector

A direction vector is obtained by subtracting the coordinates of one point from another. This vector indicates the orientation of the line in space and is fundamental in constructing any equation form of the line.

Parametric Equations

Parametric equations express each coordinate (x, y, and z) of a point on the line as functions of a single parameter, typically denoted by t. These equations are formed using a known point on the line and the direction vector, allowing the generation of all points along the line.

Symmetric Equations

Symmetric equations are derived from the parametric equations by solving each for the parameter t and equating them. They provide a way to express the line without explicitly stating the parameter and are useful for analyzing the geometric properties of the line.

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