Spherical Coordinate System Interpretation

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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. :)
 

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  • #2
haruspex
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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.
 
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  • #3
tiny-tim
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Hi Zatman! :smile:
(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°. :wink:
 
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Got it. I didn't realise "great" actually meant something here. Thanks to you both. :)
 

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