Final answer:
To determine the position vectors and distance between two planes from a control tower, trigonometry and vector calculations are used. However, without specific details such as the horizontal distance from the control tower, an exact answer cannot be provided.
Step-by-step explanation:
The question refers to a scenario where an air traffic controller is monitoring aircraft positions relative to the control tower. This situation deals with trigonometry and vector calculations, often covered in high school physics or mathematics classes. To find the position vectors of the planes, one would need to determine the horizontal and vertical components of each aircraft's position as it moves. These components are derived from the given angles of ascent and directions of movement using trigonometric functions.
To calculate the distance between the planes, first determine the position vectors and then use the distance formula in three dimensions. Since the question does not provide all the necessary details, such as the horizontal distance of the planes from the control tower, an exact answer to part (b) cannot be given without further information. However, we can assume that if the necessary distances were provided, the calculation would involve subtracting one position vector from the other and finding the magnitude of the resulting vector.