Final answer:
To find the position vectors of the Boeing 747 and Douglas DC-3 relative to the control tower, we must use trigonometry and physics principles. The frequency mentioned, related to air traffic controller communications, is not used in this calculation. After deriving the position vectors, the distance between the two aircraft is determined using the distance formula.
Step-by-step explanation:
To solve this physics problem, we have been given the task to find the position vectors of two planes relative to a control tower and calculate the distance between them. First, we must understand that the control tower frequency is not directly related to the calculation of position vectors or distances but to the communication with aircraft.
For plane Boeing 747, we will use trigonometry to determine its position vector relative to the control tower. It's climbing at a 10° incline above horizontal and moving 30° north of west. For the Douglas DC-3, also using trigonometry, as it's climbing at 5° above horizontal and moving directly west. The actual calculations would involve breaking down the movements into their respective components using the angles given and the altitudes.
After finding the position vectors of both planes, we would use the distance formula to find the distance between them at the given moment noted by the air traffic controller. This formula is derived from the Pythagorean theorem and often involves square roots and squares of the differences between corresponding components of position vectors for each plane.