Final answer:
The question pertains to calculating the position vectors of two aircraft and the distance between them using physics principles, particularly vector analysis and trigonometry.
Step-by-step explanation:
The question involves finding the position vectors of two planes and the distance between them as monitored by an air traffic controller. This requires knowledge of physics concepts including vectors and trigonometry.
(a) To calculate the position vectors of the Boeing 747 and the Douglas DC-3 relative to the control tower, one would use trigonometry to resolve their movements into horizontal and vertical components. For the Boeing climbing at 10° above the horizontal and moving 30° north of west, the position vector can be found using the sine and cosine functions applied to the climb angle and the compass heading. For the DC-3 climbing at 5° above the horizontal and cruising directly west, the position vector is simpler to calculate, requiring only the application of trigonometry to the climb angle since there is no deviation from the westward heading.
(b) The distance between the planes can be determined by first establishing the position vectors from part (a) and then using the Pythagorean theorem or vector subtraction to find the straight-line distance between the two vectors representing the positions of the two planes.