(a) Find parametric equations for the line through (5, 2, 4) that is perpendicular to the plane x - y + 2z = 3. (Use the parameter t.) (x(t), y(t), z(t)) = (
(a) Find parametric equations for the line through (5, 2, 4) that is perpendicular to the plane x - y + 2z = 3. (Use the parameter t.) (x(t), y(t), z(t)) = (
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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![### Section (a): Parametric Equations for the Line
**Problem Statement:**
Find parametric equations for the line through the point (5, 2, 4) that is perpendicular to the plane given by the equation \( x - y + 2z = 3 \). Use the parameter \( t \).
The parametric equations are of the form:
\[ (x(t), y(t), z(t)) = \left( \begin{array}{c} \ \ \ \ \ \ \end{array} \right) \]
### Section (b): Intersection Points with Coordinate Planes
**Problem Statement:**
Determine at what points this line intersects the coordinate planes.
#### Intersections:
**1. Intersection with the \( xy \)-plane:**
\[ (x(t), y(t), z(t)) = \left( \begin{array}{c} \ \ \ \ \ \ \end{array} \right) \]
**2. Intersection with the \( yz \)-plane:**
\[ (x(t), y(t), z(t)) = \left( \begin{array}{c} \ \ \ \ \ \ \end{array} \right) \]
**3. Intersection with the \( xz \)-plane:**
\[ (x(t), y(t), z(t)) = \left( \begin{array}{c} \ \ \ \ \ \ \end{array} \right) \]
**Explanation:**
This section requires identifying where the parametric line intersects the three primary coordinate planes. Each plane can be defined as one of the variables (z, x, or y) being zero:
- \( xy \)-plane: \( z = 0 \)
- \( yz \)-plane: \( x = 0 \)
- \( xz \)-plane: \( y = 0 \)
Animated diagrams or graphs should illustrate where these intersections occur, clearly indicating the points at which the line meets each plane.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fab6b79a9-c663-4f87-993a-883a678be91b%2Fd0065461-a9dd-4077-9548-dbdd2c02cd60%2F8dxy1nk_processed.png&w=3840&q=75)
Transcribed Image Text:### Section (a): Parametric Equations for the Line
**Problem Statement:**
Find parametric equations for the line through the point (5, 2, 4) that is perpendicular to the plane given by the equation \( x - y + 2z = 3 \). Use the parameter \( t \).
The parametric equations are of the form:
\[ (x(t), y(t), z(t)) = \left( \begin{array}{c} \ \ \ \ \ \ \end{array} \right) \]
### Section (b): Intersection Points with Coordinate Planes
**Problem Statement:**
Determine at what points this line intersects the coordinate planes.
#### Intersections:
**1. Intersection with the \( xy \)-plane:**
\[ (x(t), y(t), z(t)) = \left( \begin{array}{c} \ \ \ \ \ \ \end{array} \right) \]
**2. Intersection with the \( yz \)-plane:**
\[ (x(t), y(t), z(t)) = \left( \begin{array}{c} \ \ \ \ \ \ \end{array} \right) \]
**3. Intersection with the \( xz \)-plane:**
\[ (x(t), y(t), z(t)) = \left( \begin{array}{c} \ \ \ \ \ \ \end{array} \right) \]
**Explanation:**
This section requires identifying where the parametric line intersects the three primary coordinate planes. Each plane can be defined as one of the variables (z, x, or y) being zero:
- \( xy \)-plane: \( z = 0 \)
- \( yz \)-plane: \( x = 0 \)
- \( xz \)-plane: \( y = 0 \)
Animated diagrams or graphs should illustrate where these intersections occur, clearly indicating the points at which the line meets each plane.
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