Consider a rectangular coordinate system with origin at the...

Consider a rectangular coordinate system with origin at the center of the earth, z-axis through the North Pole, and x-axis through the prime-meridian. Find the rectangular coordinates of London, England (51°32'N, 0°5'W). A minute is 1/60°. Assume the earth is a sphere of radius R = 6367 km. London has coordinates (-914.56, 4341.06, 4567). Usage: To enter a point, for example (x, y, z), type "(x, y, z)".

The latitude and longitude of a point $P$ in the Northern Hemisphere are related to spherical coordinates $\rho, \theta, \phi$ as follows. We take the origin to be the center of the earth and the positive $z$ -axis to pass through the North Pole. The positive $x$ -axis passes through the point where the prime meridian (the meridian through Greenwich, England) intersects the equator. Then the latitude of $P$ is $\alpha=90^{\circ}-\phi^{\circ}$ and the longitude is $\beta=360^{\circ}-\theta^{\circ} .$ Find the great-circle distance from Los Angeles (lat. $34.06^{\circ} \mathrm{N},$ long. $118.25^{\circ} \mathrm{W}$ ) to Montréal (lat. $45.50^{\circ} \mathrm{N},$ long. $73.60^{\circ} \mathrm{W} ) .$ Take the radius of the earth to be 3960 $\mathrm{mi}$ . (A great circle is the circle of intersection of a sphere and a plane through the center of the sphere.)

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