In the lecture we learned how to find the distance between the tips of any two vectors if we know the component expressions for each of them. In this discussion we are going to apply this knowledge to find the distance and flight time between two of your favorite cities. This in fact is the arc length connecting the two cities. The arc in question is part of a big circle with the radius R of the earth, centered at the center of the earth.
If we know the angle extended by this arc, then the arc length is simply given by L=Rθ. To find θ, lets first find the straight-line distance between the two cities, then use the Law of Cosine. Follow these steps:
Study the longitude and latitude (Links to an external site.)Links to an external site.. Establish a coordinate system centered at the center of the earth, with the equatorial plane as the xy plane and the z-axis pointing to the North Pole. If we express the latitude angle as α and the longitude angle as β, write down the component expression of the position vector of any spot on the surface of the earth.
Search the web to find the values of these angles for two of your favorite destinations (such as Paris and New York). Then find the straight-line distance between them, and then the arc length separating them — the latter is the actual distance L between the two cities. (Do not put in two cities that are too close to each other, such as LA and San Diego, as the arc length is too close to the straight-line length.)
Google again to find the official data for the distance L between the two cities — Does it agree with your calculation? (You may want to look it up from this website: http://www.wolframalpha.com (Links to an external site.)Links to an external site. Just type in whatever you want to know!)
Now, search a travel website such as www.travelocity.com (Links to an external site.)Links to an external site. to find a direct flight connecting those two cities. From the type of the airplane in service, you should be able to find its cruising speed. Does that give you a decent estimation of the flight time? (Hint: the speed of the jet stream could set your calculation off. To average out this effect, take the average flight time to- and from- one city to the other.)