Final answer:
The plane's total displacement is 6.2 km, at an angle of -48 degrees north of west.
Step-by-step explanation:
To find the total displacement of the plane, we need to break down each leg of the trip into its components. First, let's consider the plane's motion at an angle of 25 degrees north of east for 6 km. Using trigonometry, we can find that the eastward component of this leg is 6*cos(25) km and the northward component is 6*sin(25) km.
Next, the plane travels at an angle of 42 degrees north of west for 4 km. The westward component of this leg is 4*cos(42) km and the northward component is 4*sin(42) km.
Finally, the plane travels west for 7 km, resulting in a displacement of -7 km eastward and 0 km northward. To find the total displacement, we add up the eastward and westward components, as well as the northward components. The final eastward displacement is 6*cos(25) km - 4*cos(42) km - 7 km = -4.39 km. The final northward displacement is 6*sin(25) km + 4*sin(42) km + 0 km = 4.71 km.
The magnitude of the total displacement is found using the Pythagorean theorem: sqrt((-4.39 km)^2 + (4.71 km)^2) = 6.2 km. The direction of the total displacement can be found using inverse tangent: atan(4.71 km / -4.39 km) = -48 degrees north of west.