Flying by Vectors: The 3D Geometry of a Flight Path
Geometry & Trigonometry
Turn two airport coordinates into 3D vectors from the centre of the Earth, and the great-circle distance between them drops out of a single dot product.
Introduction
Every point on Earth's surface can be written as a 3D position vector from the planet's centre, using latitude and longitude. Once two cities are position vectors, the angle between them (found with the scalar product) gives the great-circle distance directly — no separate formula to memorise, just vectors you already know how to work with. This is AI HL vector content: it needs 3D vectors and the scalar product, not just the coordinates themselves.
Guiding Questions
- Convert a city's latitude and longitude into a 3D position vector (with the Earth's centre as the origin). Do this for two cities of your choice.
- Use the scalar product to find the angle between the two vectors. How does this angle relate to distance along the Earth's surface?
- Derive the great-circle distance from that angle, and check it against an online flight-distance calculator.
- For three cities, does the vector method still work if the flight isn't a simple two-point hop? Try Geneva to Tokyo via Dubai.