Latitude and longitude of the most northern point

[Rocko]

How would one find the latitude and longitude of the most northern point traveling from Vancouver to New York along the shortest path (great circle)?

 

[Rocko]

I guess this one is too hard for all you wizes. Thanks anyway.

 

[Tide]

Express the locations of both cities in three dimensional cartesian coordinates, evaluate the vector cross product and do a little geometry. I'm sure you're up to it. :-)

 

[BobG]

If you're familiar with cross products and dot products, this is fairly easy.

The cross product Tide mentioned gives you a vector perpendicular to the plane. Your start point has to be your reference if you want your resulting vector to point in the right direction, in other words, your equation is:

$$v\times n=r$$
with v being Vancouver's three dimenional coordinates and n being New York's three dimenional coordinates.

Since the geocentric z axis is perpendicular to the equaorial plane and the result of your cross product is perpendicular to your great circle route, the angle between those two vectors matches the angle between your great circle route and the equatorial plane. Use the dot product to find the angle between the geocentric z axis (unit vector k) and the result of your cross product.

$$cos\theta=\frac{k \bullet r}{kr}$$
r is the result from your cross product.
k is just 0i+0j+1k
You divide the dot product by the product of the norms (this basically simplifies to the magnitude of your resulting vector)

$$\theta$$ is the angle between the great circle route and the equator. At some point, before you can stop traveling Northeast and start traveling Southeast, you have to travel due East. The latitude this happens at matches the angle between your two planes. In other words, the angle you get from your dot product is the Northernmost latitude.

 

[Rocko]

ok i will read up on dot and cross products as we haven't even touched those yet in class. I am in a geomatics course 1st year so we are just stepping into spherical trig and haven't learned about vectors yet ?? seems bizarre. I did learn about them many years ago in 1st year physics but have long forgot but i will see if i can recall any of it if i study up.

 

[Rocko]

How do you find the cartesian coordinates of vancouver?
(N49 12 00, W123 14 00) radius of Earth 6372km.