Answer
Rectangular coordinates of Sydney:
$x = - 4618.84$ km
$y = 2560.26$ km
$z = - 3562.06$ km
Rectangular coordinates of Bogotá:
$x = 1723.67$ km
$y = - 6111.69$ km
$z = 503.11$ km
Work Step by Step
1. Sydney, Australia (34$^\circ $ S, 151$^\circ $ E)
We have the latitude 34$^\circ $ S and the longitude 151$^\circ $ E.
First we find the spherical angles $\left( {\theta ,\phi } \right)$ for Sydney:
Sydney's latitude is south of the equator, so $34^\circ = \phi - 90^\circ $ and $\phi = 124^\circ $.
Since the longitude of Sydney lies to the east of prime meridian, $\theta = 151^\circ $.
So, $\left( {\theta ,\phi } \right) = \left( {151^\circ ,124^\circ } \right)$. Using the relations between rectangular and spherical coordinates:
$x = \rho \sin \phi \cos \theta $
$y = \rho \sin \phi \sin \theta $
$z = \rho \cos \phi $
where in this case $\rho = 6370$, we get
$x = 6370\sin 124^\circ \cos 151^\circ = - 4618.84$ km
$y = 6370\sin 124^\circ \sin 151^\circ = 2560.26$ km
$z = 6370\cos 124^\circ = - 3562.06$ km
2. Bogotá, Colombia (4$^\circ $32' N, 74$^\circ $ 15' W)
We have the latitude $4^\circ 32' = 4.53^\circ $ N and the longitude $74^\circ 15' = 74.25^\circ $ W.
Since the latitude of Bogotá is north of the equator, $4.53^\circ = 90^\circ - \phi $, So, $\phi = 85.47^\circ $.
Since 74.25$^\circ $ W refers to 74.25$^\circ $ in the negative $\theta$ direction, we have $\theta = 360^\circ - 74.25^\circ = 285.75^\circ $.
So, $\left( {\theta ,\phi } \right) = \left( {285.75^\circ ,85.47^\circ } \right)$. Using the relations between rectangular and spherical coordinates:
$x = \rho \sin \phi \cos \theta $
$y = \rho \sin \phi \sin \theta $
$z = \rho \cos \phi $
where in this case $\rho = 6370$, we get
$x = 6370\sin 85.47^\circ \cos 285.75^\circ = 1723.67$ km
$y = 6370\sin 85.47^\circ \sin 285.75^\circ = - 6111.69$ km
$z = 6370\cos 85.47^\circ = 503.11$ km
