Final answer:
The problem requires using trigonometry and vector analysis to calculate the position vectors of two planes from a control tower and the distance between them, involving the application of trigonometric functions and the Pythagorean theorem.
Step-by-step explanation:
The question involves finding the position vectors and the distance between two airplanes from a fixed point, which in this case is a control tower at a regional airport. The calculation of position vectors and distances involves trigonometry and vector analysis, which are topics covered in physics.
To calculate the position vectors of the planes relative to the control tower, we first need to break down their movements into components based on the given angles of climb and their directions. Then, using trigonometric functions, we calculate the horizontal and vertical components of their position vectors from their altitudes and angles above the horizontal.
After finding the position vectors, to calculate the distance between the Boeing 747 and the Douglas DC-3, we use the Pythagorean theorem or vector subtraction to find the resultant vector between the two position vectors and then calculate the magnitude (distance) of this resultant vector.