   Chapter 12.1, Problem 6E

Chapter
Section
Textbook Problem

# What does the equation y = 3 represent in ℝ3? What does z = 5 represent? What does the pair of equations y = 3, z = 5 represent? In other words, describe the set of points (x, y, z) such that y = 3 and z = 5. Illustrate with a sketch.

To determine

To explain: The representation of equation y=3 , z=5 in 3 . The representation of pair of equations y=3 and z=5 in 3 .

Explanation

3 is the three-dimensional coordinate system which contains x, y, and z-coordinates.

The equation y=3 in 3 represents the set {(x,y,z)|y=3} , which is the set of all points in 3 whose y-coordinate is 3 and x, z-coordinates are any values.

Therefore, the equation y=3 in 3 represents the vertical plane that is parallel to the xz-plane and is located 4 units to the right on the xz-plane.

The equation z=5 in 3 represents the set {(x,y,z)|z=5} , which is the set of all points in 3 whose z-coordinate is 5 and x, y-coordinates are any values.

Therefore, the equation z=5 in 3 represents the horizontal plane that is parallel to the xy-plane and is located 5 units above the xy-plane

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