# Spherical Coordinate System Interpretation

• Zatman
In summary, the particle initially travels eastward and then turns to travel northward. After doing so, it crosses the equator at longitude ±90°.

## Homework Statement

(a) Starting from a point on the equator of a sphere of radius R, a particle travels through an angle α eastward and then through an angle β along a great circle toward the north pole. If the initial position is taken to correspond to x = R, y = 0, z = 0, show that its final coordinates are (Rcosαcosβ, Rsinαcosβ, Rsinβ).

(b) Find the coordinates of the final position of the same particle if it first travels through an angle α northward, then changes course by 90° and travels through an angle β along a great circle that starts out eastward.

2. The attempt at a solution
This is basically geometry, and I've shown part (a) correctly via a diagram. I don't seem to get the correct answer for part (b) and I suspect I am not interpreting the question correctly.

Attached is a somewhat crude (though the clearest I can produce!) diagram showing how I see the situation for (a) and (b). I haven't added my annotations for my answer for clarity's sake. I get the following for (b):

x = Rcosαcosβ
y = Rcosαsinβ
z = Rsinα

If someone could tell me whether or not I am interpreting the information correctly (and hence whether it actually is my geometric analysis) I would very much appreciate it. :)

#### Attachments

• diagrams.jpg
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Your diagram for b is wrong. After changing course it travels on a great circle. That is not a circle parallel to the equator.

1 person
Hi Zatman!
Zatman said:
(b) Find the coordinates of the final position of the same particle if it first travels through an angle α northward, then changes course by 90° and travels through an angle β along a great circle that starts out eastward.

z = Rsinα

No, that's still on the (small) circle of latitude α

the great circle dips down towards the equator, crossing it at longitude ±90°.

1 person
Got it. I didn't realize "great" actually meant something here. Thanks to you both. :)

## 1. What is a spherical coordinate system?

A spherical coordinate system is a three-dimensional coordinate system used to locate points on a sphere. It is defined by three coordinates: radius (r), inclination (θ), and azimuth (φ).

## 2. How is a point represented in spherical coordinates?

In spherical coordinates, a point is represented by the distance from the origin (r), the angle between the positive z-axis and the line connecting the point to the origin (θ), and the angle between the positive x-axis and the projection of the point onto the xy-plane (φ).

## 3. What are the advantages of using a spherical coordinate system?

A spherical coordinate system is particularly useful for representing points on a sphere or points in three-dimensional space that involve angles. It is also helpful in solving problems involving symmetry, such as in physics and engineering.

## 4. How is the spherical coordinate system related to the Cartesian coordinate system?

The spherical coordinate system is related to the Cartesian coordinate system through the following equations:
x = r sinθ cosφ
y = r sinθ sinφ
z = r cosθ
where r is the distance from the origin, θ is the inclination angle, and φ is the azimuth angle.

## 5. Can a point have more than one representation in spherical coordinates?

Yes, a point can have more than one representation in spherical coordinates. For example, a point with the coordinates (r, θ, φ) will have the same representation as a point with coordinates (r, π - θ, φ + π) or (r, θ + 2π, φ). Therefore, it is important to specify the range of values for each coordinate to avoid confusion.