Final answer:
To find the position vectors and distance between two Planes from a control tower, use trigonometry to resolve each plane's movement into components and apply the distance formula.
Step-by-step explanation:
The student is asking how to determine the position vectors of two aircraft relative to a control tower and the distance between them at a given moment.
To solve this, one must use trigonometry to decompose the movement of the planes into horizontal and vertical components and then find the vectors representing the position of each aircraft relative to the control tower. The next step involves calculating the distance between these two vectors to find the distance between the planes.
For the Boeing 747, climbing at 10° above the horizontal and moving 30° north of west at an altitude of 2500 m, we can break down the movement into east-west and north-south components using trigonometric functions.
Similarly, for the DC-3, climbing at 5° above the horizontal and moving directly west at an altitude of 3000 m, the components will only be westward and upward, as there is no north-south component. After determining the position vectors, we use the distance formula rooted in the Pythagorean theorem to calculate the distance between the planes.