Problem 67 - Thomas Calculus

Question

Find all polar coordinates of the origin.

Step-by-Step Solution

Answer

The polar coordinates of the origin are \((0, \theta)\), where \(\theta\) is any real number.

Step 1: Understanding Polar Coordinates

Polar coordinates of a point are represented as \((r, \theta)\), where \(r\) is the radial distance from the origin, and \(\theta\) is the angle measured from the positive x-axis in a counter-clockwise direction. For the origin, the radial distance \(r\) is 0.

Step 2: Assign Radial Distance

Since the origin is the point where the x and y coordinates are both 0, in polar coordinates, this means the radial distance \(r\) is also 0. Thus, one part of the polar coordinate of the origin is \(r = 0\).

Step 3: Determine the Angle

In polar coordinates, the angle \(\theta\) can be any real number when \(r = 0\) because the angle component becomes irrelevant when the radial distance is zero.

Step 4: List All Possible Polar Coordinates

Combining the determined \(r\) and \(\theta\), the polar coordinates of the origin can be written as \((0, \theta)\), where \(\theta\) can be any real number. This means there are infinitely many representations of the origin in polar coordinates.

Key Concepts

Radial Distance

Radial distance in polar coordinates is denoted by the letter \(r\). It represents the straight-line distance between the origin of the polar coordinate system and the specific point in question. It is akin to measuring the length of a radius in a circle, but for any given point on the coordinate plane.

• When \(r\) is zero, it means the point is exactly at the origin.
• \(r\) is a non-negative value, indicating how far you are from the center of your coordinate system.

The radial distance is crucial because it determines where a point is located relative to the origin. In cases such as the origin itself, \(r\) is particularly simple as it equals zero, signaling no distance at all from the origin. Understanding radial distance simplifies mapping various points, especially when converting between different coordinate systems, like Cartesian and polar.

Angle Measurement

Angle measurement in polar coordinates is symbolized by \(\theta\). It signifies the angle formed from the positive x-axis to the line connecting the origin to the point. This angular component adds directional information to a point's location, making it a core element of polar coordinates.

• Angle \(\theta\) is typically measured in radians for mathematical convenience, but degrees can also be used.
• The angle is measured counter-clockwise from the positive x-axis.

When the radial distance \(r = 0\), as at the origin, angle \(\theta\) can be any value because the point doesn’t move away from the origin regardless of the angle considered. This means that although \(r\) remains defined, \(\theta\) does not alter the point's position. This provides a fascinating property of the origin in polar coordinates: its infinite angular possibilities.

Origin in Polar Coordinates

The origin, a key reference point in any coordinate system, translates interestingly in polar coordinates. Here, it is represented as \((0, \theta)\), with \(r = 0\) and \(\theta\) as any real number. This representation emphasizes:

• The radial distance at the origin is always zero, simplifying its representation in the polar system.
• The angle \(\theta\), while flexible, does not change the actual position of the origin.

This results in a unique representation: an infinite number of angles can describe the origin, making it distinct from any other point in the polar system. Understanding this helps clarify not only the conceptual translation between Cartesian and polar systems but also highlights the flexibility and adaptability of polar coordinates for applications involving circular symmetry or periodic functions.