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What does the equation y = 8 represent in R³? O a point a line a plane a circle What does z = 6 represent? O a point O a line a plane a circle What does the pair of equations y = 8, z = 6 represent? In other words, describe the set of points (x, y, z) such that y = 8 and z = 6. O a point a line a plane O a circle Illustrate with a sketch.

What does the equation y = 8 represent in R³? O a point a line a plane a circle What does z = 6 represent? O a point O a line a plane a circle What does the pair of equations y = 8, z = 6 represent? In other words, describe the set of points (x, y, z) such that y = 8 and z = 6. O a point a line a plane O a circle Illustrate with a sketch.

Calculus: Early Transcendentals
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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## Mathematical Representation in ℝ³ ### 1. What does the equation \( y = 8 \) represent in \( \mathbb{R}^3 \)? - \( \quad \) \( \circ \) a point - \( \quad \) \( \circ \) a line - \( \quad \) \( \bullet \) a plane - \( \quad \) \( \circ \) a circle The equation \( y = 8 \) represents a plane in three-dimensional space \( \mathbb{R}^3 \). ### 2. What does \( z = 6 \) represent? - \( \quad \) \( \circ \) a point - \( \quad \) \( \circ \) a line - \( \quad \) \( \bullet \) a plane - \( \quad \) \( \circ \) a circle The equation \( z = 6 \) represents a plane in three-dimensional space \( \mathbb{R}^3 \). ### 3. What does the pair of equations \( y = 8, \, z = 6 \) represent? In other words, describe the set of points \( (x, y, z) \) such that \( y = 8 \) and \( z = 6 \). - \( \quad \) \( \circ \) a point - \( \quad \) \( \bullet \) a line - \( \quad \) \( \circ \) a plane - \( \quad \) \( \circ \) a circle The pair of equations \( y = 8 \) and \( z = 6 \) represents a line in three-dimensional space \( \mathbb{R}^3 \). ### Explanation with a Sketch To illustrate this, one can sketch the \( \mathbb{R}^3 \) space: - Draw the \( x \)-, \( y \)-, and \( z \)-axes. - Show the plane \( y = 8 \) (a plane parallel to the \( xz \)-plane positioned at \( y = 8 \)). - Show the plane \( z = 6 \) (a plane parallel to the \( xy \)-plane positioned at \( z = 6 \)). - The intersection of these two planes will be a line along the \( x \)-axis
Transcribed Image Text:## Mathematical Representation in ℝ³ ### 1. What does the equation \( y = 8 \) represent in \( \mathbb{R}^3 \)? - \( \quad \) \( \circ \) a point - \( \quad \) \( \circ \) a line - \( \quad \) \( \bullet \) a plane - \( \quad \) \( \circ \) a circle The equation \( y = 8 \) represents a plane in three-dimensional space \( \mathbb{R}^3 \). ### 2. What does \( z = 6 \) represent? - \( \quad \) \( \circ \) a point - \( \quad \) \( \circ \) a line - \( \quad \) \( \bullet \) a plane - \( \quad \) \( \circ \) a circle The equation \( z = 6 \) represents a plane in three-dimensional space \( \mathbb{R}^3 \). ### 3. What does the pair of equations \( y = 8, \, z = 6 \) represent? In other words, describe the set of points \( (x, y, z) \) such that \( y = 8 \) and \( z = 6 \). - \( \quad \) \( \circ \) a point - \( \quad \) \( \bullet \) a line - \( \quad \) \( \circ \) a plane - \( \quad \) \( \circ \) a circle The pair of equations \( y = 8 \) and \( z = 6 \) represents a line in three-dimensional space \( \mathbb{R}^3 \). ### Explanation with a Sketch To illustrate this, one can sketch the \( \mathbb{R}^3 \) space: - Draw the \( x \)-, \( y \)-, and \( z \)-axes. - Show the plane \( y = 8 \) (a plane parallel to the \( xz \)-plane positioned at \( y = 8 \)). - Show the plane \( z = 6 \) (a plane parallel to the \( xy \)-plane positioned at \( z = 6 \)). - The intersection of these two planes will be a line along the \( x \)-axis
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