Question

For constants \(a, b,\) and \(c,\) describe the graphs of the equa tions \(r=a, \theta=b,\) and \(z=c\) in cylindrical coordinates.

Short answer

The graph of \(r=a\) is a cylinder of radius \(a\) along the z-axis. The graph of \(\theta=b\) is a half-plane that has rotated an angle \(b\) from the positive x-axis. The graph of \(z=c\) is a horizontal plane parallel to the xy-plane at a height \(c\).

Step-by-step solution

Visualizing \(r=a\)

The graph of \(r=a\) depicts a cylinder of radius \(a\) centered on the z-axis. The height (\(z\)) and angle (\(\theta\)) can take any values, but the distance from the z-axis is always \(a\). So, it's a circle with radius \(a\) that has been extended indefinitely in the positive and negative z-directions.

Visualizing \(\theta=b\)

The graph of \(\theta=b\) is a half-plane starting from the z-axis and rotating an angle of \(b\) in the counterclockwise direction. The radius (\(r\)) and the height (\(z\)) can take any values, which means this plane extends indefinitely in the radial and vertical directions.

Visualizing \(z=c\)

The graph of \(z=c\) represents a horizontal plane that is \(c\) units above the xy-plane if \(c>0\), \(c\) units below the xy-plane if \(c<0\), or coincides with the xy-plane if \(c=0\). The plane extends indefinitely in the radial and angular directions, because \(r\) and \(\theta\) can be any values.