A coordinate system on the earth is established so that the...

Name: a coordinate system on the earth is established so that the north pole is at 003960 the equator lies in the xy plane the prime meridian including greenwich england lies in the x z plane and 87418
Uploaded: 2020-06-21T20:56:36-08:00
Duration: 7 min 30 s
Channel: Christian Otero
Description: a coordinate system on the earth is established so that the north pole is at 003960 the equator lies in the xy plane the prime meridian including greenwich england lies in the x z plane and 87418

A coordinate system on the Earth is established so that the North Pole is at (0,0,3960); the equator lies in the xy-plane; the prime meridian (including Greenwich, England) lies in the x-z-plane; and the y-axis points toward the Indian Ocean. Find the x, y, and z coordinates of a town that has latitude 55 degrees north and longitude 75 degrees west.

Transcript

00:01 All right, so what we want to do here is we have two coordinates on the earth.

00:07 Those are los angeles and montereo.

00:11 So we have these two latitude and longitude sets and these corresponding equations that relate our latitude and alpha and our longitude beta to our spherical coordinates.

00:31 Phi and theta.

00:35 So what we want to do is go ahead and try to find the great circle distance.

00:38 And if you're not sure what a great circle is, when we have a sphere like this, a great circle is a circle that's basically in the plane with the center.

00:56 So it's this kind of circle, and then the great circle distance is the distance between two points along this circle.

01:05 So it's sort of like the arc length on this circle.

01:08 We know our final formula for this arc length should be, so we'll call it the for distance, it should be some, the radius or row times whatever angle is between these points.

01:28 So we have this, let me write in black, so we have this angle that i'm going to call x.

01:39 So we call this angle x.

01:43 So if we use radians, we just have our distance equal to the radius of this sphere, or in this case the great circle, times the angle x.

01:53 And if we do the solid degrees, we have this distance is equal to, we know that this angle x should just be pi times degrees over 180.

02:05 So we have row pi over 180 times x degrees times x degrees.

02:13 So since we're already using degrees, we're not going to have, we're not going to transfer anymore.

02:19 We're just going to do this.

02:20 So what this means is we want to find the angle between this latitude, these two latitudes and longitude sets that we have for la and montreal.

02:29 So what we want to do, what i'll do is go ahead and convert everything to rectangular coordinates.

02:37 And then once we have rectangular coordinates, we can use our knowledge of vectors and dot products to find angles between two points.

02:45 In 3d space and then use that to find our great circle distance.

02:50 So our fundamental relationships for this problem are the following.

02:55 X is equal to row cosine theta sine phi, y is equal to row sine theta sine phi, and z is equal to row sine phi, i mean row cosine phi.

03:10 Now we're given that the radius of the earth is 3 ,960 miles and these relationships, between phi in longitude, phi in latitude and theta longitude.

03:26 If we basically switch the order of these two variables, we get that our theta is equal to 360 minus beta and our phi is equal to 90 minus alpha.

03:36 So we can go ahead and use our equations to find our coordinates in rectangular space...

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