An ornithologist is in the field studying a formation of geese and is 180 m south, and 385 m west, of where she parked her car.
Let u be the vector representing the distance between the ornithologist and her car, and let v be the vector representing the straight line distance from the ornithologist to the formation.
If the lead goose flies over her car at an altitude 319 m
find the angle btw u and v using cos `theta` = u.v/|u||v|
Verify result using right angle trig.
An ornithologist is in the field studying the formation of geese. She is 180 m south, and 385 m west of where she parked her car. The lead goose flies over her car at an altitude of 319 m. `vec u ` represents the vector from the ornithologist to her car. `vec v ` represents the vector from the ornithologist to the lead goose.
Let the point where the ornithologist is standing be (0,0,0). The vector `vec u` is [180, 385, 0] and the vector `vec v` is [180, 385, 319]
The magnitude of `vec v` is `sqrt(180^2 + 385^2 + 319^2)` = 531.4
The magnitude of `vec u` is `sqrt(180^2 + 385^2)` = 425.
The angle `theta` between `vec u` and `vec v` is defined as `cos theta = (180^2 + 385^2 + 0)/(531.4*425)`
=> `theta = cos^-1(2125/2657)` = 36.89 degrees
Using the right triangle formed the angle is `tan^-1(319/425)` = 36.89 degrees
The angle between the vectors is verified to be 36.89 degree.