Question

Why are points on the \(z\) -axis not determined uniquely when using cylindrical and spherical coordinates?

Short answer

In cylindrical and spherical coordinates, points on the \( z \)-axis have arbitrary \( \theta \) because there's no radial component in the \( xy \)-plane.

Step-by-step solution

Understand Cylindrical Coordinates

Cylindrical coordinates consist of three values: \( (r, \theta, z) \). Here, \( r \) is the radial distance from the origin to the projection of the point in the \( xy \)-plane, \( \theta \) is the angle from the positive \( x \)-axis to this projection, and \( z \) is the height along the \( z \)-axis.

Identify Issue with Cylindrical Coordinates

On the \( z \)-axis, all points have the form \((0, \theta, z)\) because the distance from the origin to the projection in the \( xy \)-plane is zero (\( r = 0 \)). The angle \( \theta \) is undefined or arbitrary because there are infinite orientations around the \( z \)-axis where \( r = 0 \).

Understand Spherical Coordinates

Spherical coordinates consist of three values: \( (\rho, \theta, \phi) \). Here, \( \rho \) is the radial distance from the origin to the point, \( \theta \) is the angle from the positive \( x \)-axis, and \( \phi \) is the angle from the positive \( z \)-axis.

Identify Issue with Spherical Coordinates

For points on the \( z \)-axis, \( \rho \) equals \( z \) because there is no radial component in the \( xy \)-plane. The angle \( \theta \) is arbitrary or undefined in this case because any rotation around the \( z \)-axis does not affect the position of a point that is directly on the \( z \)-axis.

Explain the Lack of Unique Specification

In both cylindrical and spherical coordinates, the issue of undefined or arbitrary \( \theta \) occurs because the point only varies along the \( z \)-axis and has no observable radial component in the \( xy \)-plane that would affect \( \theta \). This makes points on the \( z \)-axis not uniquely determined by the angle.

Key concepts

Cylindrical Coordinates

Cylindrical coordinates are a way to express the location of a point in three-dimensional space using the values \((r, \theta, z)\). These coordinates are akin to polar coordinates but extended to three dimensions. The component \(r\) represents the radial distance from the origin to the point's projection on the \(xy\)-plane.
\(\theta\) is the azimuthal angle that describes the direction of the radial distance with respect to the positive \(x\)-axis, essentially indicating rotation around the \(z\)-axis.
Finally, \(z\) accounts for the height or depth along the \(z\)-axis itself. This is somewhat intuitive as it vertically aligns with the Cartesian coordinate system's \(z\) value.
For points specifically located on the \(z\)-axis, the radial distance \(r = 0\), which results in \(\theta\) being undefined. The reason for this is simple: a point directly along the \(z\)-axis does not extend horizontally in the \(xy\)-plane, thus removing any directional properties that \(\theta\) would indicate.

Spherical Coordinates

Spherical coordinates offer another fascinating system for pinpointing a point in 3D space, utilizing \((\rho, \theta, \phi)\). Here, \(\rho\) symbolizes the radial distance from the origin directly to the point in 3D. It gives us a measure of how far the point is from the origin.
\(\theta\) is like the version from cylindrical coordinates, measuring the angle between the projection of the point into the \(xy\)-plane and the positive \(x\)-axis.
However, \(\phi\) is unique to spherical coordinates and denotes the angle between the positive \(z\)-axis and the line connecting the origin to the point.
On the \(z\)-axis, the radial distance \(\rho\) becomes simply the \(z\)-coordinate value itself, causing \(\theta\) to become irrelevant or undefined, as every direction around the \(z\)-axis appears identical from that specific position.

Z-Axis

The \(z\)-axis stands as one of the three axes in a three-dimensional coordinate system. It is the vertical line that supports the height component in both cylindrical and spherical systems.
In cylindrical coordinates, the height \(z\) represents the vertical positioning, which does not contribute to the radial components or directional angles if a point resides directly on this axis.
Likewise, in spherical coordinates, the height along the \(z\)-axis translates directly to \(\rho\) when no other dimensional extension exists in the \(xy\)-plane. This purely vertical alignment is why angles defining directions like \(\theta\) become undefined here. Points on the \(z\)-axis reflect their vertical status, negating changes in \(xy\)-plane directionality.

Undefined Angle

An undefined angle in the context of cylindrical and spherical coordinates typically refers to the azimuthal angle \(\theta\) when points lie directly on the \(z\)-axis. This occurs naturally because there's no radial distance extending into the \(xy\)-plane—\(r = 0\), meaning that the point is lodged vertically and possesses no perceptible direction in terms of rotation around the \(z\)-axis.
Such a scenario leads to ambiguity since for any point directly up the \(z\)-axis, the rotational orientation doesn't change the position because there is no horizontal displacement for \(\theta\) to express. Thus, we say \(\theta\) is undefined or arbitrary, lacking unique determination.

Radial Distance

Radial distance is a crucial concept in both cylindrical and spherical coordinates. In cylindrical coordinates, radial distance \(r\) measures how far a point extends from the origin in the \(xy\)-plane. It's a horizontal measure used to express the initial reach of a point before considering height.

• When \(r = 0\), it implies no extension horizontally, placing the point somewhere along the \(z\)-axis.
  

In spherical coordinates, the radial distance component \(\rho\) presents the direct line measure from the origin to the point in a complete three-dimensional sense.
Just like \(r\), when \(\rho\) effectively captures only the length along the \(z\)-axis (i.e., the point aligns with the \(z\)-axis), the directional component \(\theta\) becomes unnecessary. Hence, radial distance defines a major part of how point positions create complication, especially when combined with vertical pure \(z\)-axis alignment.