Final answer:
The airplane's velocity relative to the air is (68.76 mph, 296.80 mph). The wind's velocity is (20 mph, 34.64 mph). The sum of the vectors is (88.76 mph, 331.44 mph). The true speed of the airplane is approximately 343.72 mph and its direction is 76.81°.
Step-by-step explanation:
Part A:
To write the vectors in linear form, we will break each vector into its horizontal and vertical components. The airplane's velocity relative to the air is 300 mph at a direction of 100°. The horizontal component of the vector is given by 300 mph * cos(100°), which is approximately 68.76 mph. The vertical component is given by 300 mph * sin(100°), which is approximately 296.80 mph. So, the linear form of the airplane's velocity relative to the air is (68.76 mph, 296.80 mph).
Part B:
To find the sum of the vectors, we will add their respective horizontal and vertical components. The horizontal component of the sum is 68.76 mph + 20 mph, which is approximately 88.76 mph. The vertical component is 296.80 mph + 34.64 mph, which is approximately 331.44 mph. So, the sum of the vectors is (88.76 mph, 331.44 mph).
Part C:
To find the true speed and direction of the airplane, we will calculate the magnitude and direction of the sum of the vectors. The magnitude of the sum is given by the Pythagorean theorem: sqrt((88.76 mph)^2 + (331.44 mph)^2), which is approximately 343.72 mph. The direction of the sum is given by the inverse tangent of the vertical component divided by the horizontal component: atan(331.44 mph / 88.76 mph), which is approximately 76.81°. So, the true speed of the airplane is 343.72 mph and its direction is 76.81°.