y = -1+t, and z = 2+3

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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**Question 4:** 

Find the point of intersection (if any) of the line with parametric equations \( x = 1 + 2t \), \( y = -1 + t \), and \( z = 2 + 3t \), and the plane \( 2x + 3y - 4z = 6 \).

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**Explanation for Educational Website:**

This question involves finding the intersection between a line and a plane in three-dimensional space. The line is described by parametric equations, which express the coordinates \(x\), \(y\), and \(z\) in terms of a parameter \(t\). The plane is given by a linear equation involving \(x\), \(y\), and \(z\). Solving this requires substituting the parametric equations into the plane's equation to find the value of \(t\) at which the intersection occurs, if it exists.
Transcribed Image Text:**Question 4:** Find the point of intersection (if any) of the line with parametric equations \( x = 1 + 2t \), \( y = -1 + t \), and \( z = 2 + 3t \), and the plane \( 2x + 3y - 4z = 6 \). --- **Explanation for Educational Website:** This question involves finding the intersection between a line and a plane in three-dimensional space. The line is described by parametric equations, which express the coordinates \(x\), \(y\), and \(z\) in terms of a parameter \(t\). The plane is given by a linear equation involving \(x\), \(y\), and \(z\). Solving this requires substituting the parametric equations into the plane's equation to find the value of \(t\) at which the intersection occurs, if it exists.
Expert Solution
Step 1

Consider the parametric equation:

x=1+2ty=-1+tz=2+3t

Consider the given plane:

2x+3y-4z=6

Now plug-in the parametric in the plane:

21+2t+3-1+t-42+3t=62+4t-3+3t-8-12t=6-5t-9=6-5t=15t=-3

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