Los Angeles is located at 34.05 degree North latitude and 118.24 degree West longitude, and Rio de Janeiro, Brazil, is located at 22.90 degree South latitude and 43.23 degree West longitude (see figure). Assume that Earth is spherical and has a radius of 4000 miles. Find the spherical coordinates for the location of each city. Find the rectangular coordinates for the location of each city. Find the angle (in radians) between the vectors from the center of Earth to the two cities. Find the great-circle distance s between the cities. Repeat parts (a)-(d) for the cities of Boston, located at 42.36 degree North latitude and 71.06 degree West longitude, and Honolulu, located at 21.31 degree North latitude and 157.86 degree West longitude.
The Correct Answer and Explanation is:
Final Answers
(a) Spherical Coordinates
Each spherical coordinate is (r,θ,ϕ)(r, \theta, \phi), where:
- r=4000r = 4000 miles
- θ\theta is longitude converted to radians
- ϕ=90∘−latitude\phi = 90^\circ – \text{latitude}, also in radians
Los Angeles:
(4000,4.2195,0.9765)(4000, 4.2195, 0.9765)
Rio de Janeiro:
(4000,5.5287,1.9705)(4000, 5.5287, 1.9705)
Boston:
(4000,5.0430,0.8315)(4000, 5.0430, 0.8315)
Honolulu:
(4000,3.5280,1.1989)(4000, 3.5280, 1.1989)
(b) Rectangular Coordinates
Converted from spherical to (x,y,z)(x, y, z):
Los Angeles:
(−1568.17,−2919.72,2239.66)(-1568.17, -2919.72, 2239.66)
Rio de Janeiro:
(2684.74,−2523.79,−1556.50)(2684.74, -2523.79, -1556.50)
Boston:
(959.36,−2795.68,2695.15)(959.36, -2795.68, 2695.15)
Honolulu:
(−3451.74,−1404.41,1453.66)(-3451.74, -1404.41, 1453.66)
(c) Angle Between Vectors (Radians)
Los Angeles – Rio de Janeiro:
≈ 1.5913 radians
Boston – Honolulu:
≈ 1.2836 radians
(d) Great-Circle Distance (miles)
Los Angeles – Rio de Janeiro:
≈ 6365.04 miles
Boston – Honolulu:
≈ 5134.29 miles
Explanation
To find the distances and directions between cities on Earth, we use spherical and rectangular coordinate systems based on Earth’s approximate spherical shape. First, the spherical coordinates (r,θ,ϕ)(r, \theta, \phi) require converting latitude and longitude:
- Radius r=4000r = 4000 miles (given)
- ϕ=90∘−latitude\phi = 90^\circ – \text{latitude} converts geographic latitude to the angle from the North Pole
- θ=longitude\theta = \text{longitude} in radians (converted to positive angles from the prime meridian eastward)
Using trigonometric formulas, these spherical coordinates are converted to rectangular (Cartesian) coordinates (x,y,z)(x, y, z) for vector calculations:
- x=rsin(ϕ)cos(θ)x = r \sin(\phi) \cos(\theta)
- y=rsin(ϕ)sin(θ)y = r \sin(\phi) \sin(\theta)
- z=rcos(ϕ)z = r \cos(\phi)
Next, we find the angle between the position vectors of two cities using the dot product: cos(θ)=a⃗⋅b⃗∣a⃗∣∣b⃗∣\cos(\theta) = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}
Taking the arccos of the result gives the central angle in radians.
Finally, the great-circle distance, or shortest path along Earth’s surface, is: s=r⋅θs = r \cdot \theta
This process was applied to two city pairs:
- Los Angeles to Rio de Janeiro: ~6365 miles apart
- Boston to Honolulu: ~5134 miles apart
This method accurately models real-world geography and is used in aviation, navigation, and global communications.
