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Chapter 51: Problem 900

Describe the geometric shape determined by the graph of the equation \(\mathrm{x}^{2}+\mathrm{y}^{2}=1\) when plotted on a 3 -dimensional system of rectangular coordinates.

Short Answer

Expert verified
The given equation \(x^2 + y^2 = 1\) represents a circle with radius 1, centered at the origin in a 2-dimensional plane (x, y). When plotted in a 3-dimensional system (x, y, z), this circle remains the same at every z-coordinate, resulting in a cylinder with radius 1, centered along the z-axis and extending indefinitely in both positive and negative z-directions. The circular cross-sections parallel to the xy-plane remain unchanged throughout the entire length of the cylinder.

Step by step solution

01

Identify the given equation, and analyze it in a 2D plane

The given equation is \(x^2 + y^2 = 1\). This equation represents a circle with radius 1, centered at the origin, when plotted in a 2-dimensional plane (x, y).
02

Plot the equation in a 3-dimensional system (x, y, and z-axis)

When plotting this equation in a 3-dimensional system (x, y, z), we notice that the circle lies on the xy-plane, as the z-coordinate is not mentioned in the equation. This means that the shape remains the same, regardless of the z-coordinate.
03

Describe the geometrical shape in a 3D system

The resulting geometrical shape in a 3D system is a cylinder with radius 1, centered along the z-axis, and extending indefinitely in both the positive and negative z-directions. The z-axis passes through the center of the circle, and the circular cross-sections parallel to the xy-plane remain unchanged throughout the entire length of the cylinder.

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