Question

Find parametric equations for the line through $P_{1}$ and $P_{2}$ and also for the line segment joining those points. $$ \begin{array}{llll}{\text { (a) } P_{1}(3,-2),} & {P_{2}(5,1)} & {\text { (b) } P_{1}(5,-2,1),} & {P_{2}(2,4,2)}\end{array} $$

Name: Problem 3, Chapter 11: Calculus Early Transcendentals
Uploaded: 2020-09-05T11:58:00-08:00
Duration: 2 min 49 s
Channel: Zachary Mitchell
Description: Find parametric equations for the line through P1 and P2 and also for the line segment joining those points. (a) P1(3,-2), P2(5,1) (b) P1(5,-2,1), P2(2,4,2)

   Find parametric equations for the line through $P_{1}$ and $P_{2}$ and also for the line segment joining those points.
$$
\begin{array}{llll}{\text { (a) } P_{1}(3,-2),} & {P_{2}(5,1)} & {\text { (b) } P_{1}(5,-2,1),} & {P_{2}(2,4,2)}\end{array}
$$

Calculus Early Transcendentals

Howard Anton, Irl… 9th Edition

Chapter 11, Problem 3

Instant Answer

Solved by Expert Zachary Mitchell

09/05/2020

Step 1

This is done by subtracting the coordinates of $P_{1}$ from the coordinates of $P_{2}$. For part (a), the vector from $P_{1}$ to $P_{2}$ is $(5-3, 1-(-2)) = (2,3)$. For part (b), the vector from $P_{1}$ to $P_{2}$ is $(2-5, 4-(-2), 2-1) = (-3,6,1)$. Show more…

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Find parametric equations for the line through $P_{1}$ and $P_{2}$ and also for the line segment joining those points. $$ \begin{array}{llll}{\text { (a) } P_{1}(3,-2),} & {P_{2}(5,1)} & {\text { (b) } P_{1}(5,-2,1),} & {P_{2}(2,4,2)}\end{array} $$

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Key Concepts

Direction Vectors

A direction vector indicates the orientation of a line in space and is obtained by subtracting the coordinates of one point from another. It serves as the 'driving force' in the parametric representation, showing how to move from one point to any other along the line.

Line Segment Parameterization

Line segment parameterization involves the formulation of parametric equations with a restricted parameter domain, often the interval [0, 1]. This ensures that the equations only represent the finite part of the line that connects the two endpoints.

Parametric Equations

Parametric equations express the coordinates of points on a line as functions of one or more variables (parameters). This method features an initial point and a direction vector, allowing the description of every point along the line via an equation that combines these elements.

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