Final answer:
The question involves using trigonometry and vector analysis methods to find the position vectors of two aircraft in relation to a control tower and then calculating the distance between them. This requires drawing the vectors to scale, measuring magnitudes and angles with a ruler and protractor, respectively, and applying vector addition or subtraction.
Step-by-step explanation:
When given a radar vector by ATC, air traffic controllers and pilots must be proficient in the concepts of vectors and navigation.
To find the position vectors of the planes relative to the control tower, you would need to apply trigonometry and vector analysis skills. In the scenario where a Boeing 747 is climbing at 10° above the horizontal and moving 30° north of west, and a Douglas DC-3 is climbing at 5° above the horizontal and cruising directly west, you must establish a coordinate system, use a ruler and protractor to draw and measure vectors, and apply principles of vector addition or subtraction to find the resultant vectors that represent each plane's position relative to the control tower.
Once the vectors are drawn to scale and their magnitudes and directions are determined, you can calculate the distance between the planes using the Pythagorean theorem or appropriate vector algebra.
To measure the magnitude and direction of the resultant vector (R) representing the position of a plane relative to the control tower, you use both a ruler and protractor.
You'll measure the length of the vector with the ruler, converting it back to meters using the scale provided, and measure the angle of direction with the protractor, taking care to align the protractor correctly depending on the vector's position relative to the horizontal or vertical axes.