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Rectangular coordinates are a common coordinate system used in algebra and geometry to describe the position of points in a plane or 3D space. Each point is identified by an ordered pair

• \((x, y)\) in 2D
• \((x, y, z)\) in 3D

where \(x\), \(y\), and \(z\) represent the distances from the origin along the x-axis, y-axis, and z-axis respectively. This system is intuitive because it corresponds directly with Cartesian coordinates, which are visualized as a grid where each point is defined by its horizontal and vertical displacement from a reference point called the origin.

When dealing with equations in rectangular coordinates, such as \(x^2 + y^2 = 16\), they describe geometrical objects in space. Here, \(x^2 + y^2 = 16\) identifies a circle in the plane with a radius of 4 centered at the origin. In three dimensions, this equation specifies a cylindrical surface with an infinite extent along the z-axis.

In three-dimensional space, a cylinder is a surface formed by tracing a straight line that moves parallel to a fixed line. A simple example of a cylinder is formed by a vertical line moving around a circle while maintaining its orientation parallel to the z-axis.

The formula \(x^2 + y^2 = r^2\) represents a cylindrical shape especially when extended into 3D, where:

• \(r\) is the radius of the cylinder's base
• \((x, y)\) are the position coordinates in the plane of the circle

The z-variable is free, meaning the shape extends infinitely up and down along the z-axis. This concept makes it easier to visualize how cylindrical structures behave in space, such as pipelines or towers.

In our case, "the cylinder \(x^2 + y^2 = 16\)" represents a vertical cylinder of radius 4 that extends along the z direction without bounds.

Converting between different coordinate systems is a fundamental skill, especially when solving problems across various mathematical fields. Cylindrical coordinates are expressed using:

• \(r\): the radial distance from the z-axis
• \(\theta\): the angle from the positive x-axis
• \(z\): the height above the xy-plane

Converting from cylindrical to rectangular coordinates involves using the equations:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)
• \(z = z\)

For instance, with \(r = 4\) and using the identities for coordinate conversion, we find that the surface is defined as \(\sqrt{x^2 + y^2} = 4\). Solving this gives \(x^2 + y^2 = 16\), showing that the surface is indeed a cylinder in rectangular coordinates.

Graphing surfaces in 3D can be a bit more complex than dealing with lines or curves in two dimensions. A graph of a 3D surface shows not only the shape of the surface but also its orientation and how it extends through space.

When graphing the equation \(x^2 + y^2 = 16\) in three dimensions, one identifies features such as:

• The circular cross-section with a radius of 4 in planes parallel to the xy-plane
• The infinite extension along the z-axis, indicating a vertical cylinder

To visualize such surfaces, it's crucial to consider how every point fits into the equation and how the overall shape behaves in three-dimensional space. Graphing tools can aid in illustrating these surfaces and gaining a better sense of their volume and contour.