Final answer:
To find the resultant vector of the plane's velocity and the wind's velocity, split each velocity into its x and y components. Add the x and y components separately, then use the Pythagorean theorem and inverse tangent to find the magnitude and direction of the resultant vector.
Step-by-step explanation:
To find the resultant vector, we need to add the vectors of the plane's velocity and the wind's velocity. We can break down each velocity into its x and y components. The plane's velocity is 81.4 m/s at an angle of 34.6° S of W. This can be split into x = 81.4 * cos(34.6°) and y = -81.4 * sin(34.6°). The wind's velocity is 34.4 m/s at an angle of 12.5° E of N. This can be split into x = 34.4 * sin(12.5°) and y = 34.4 * cos(12.5°). Now we can add the x and y components of the plane's velocity and wind's velocity separately to get the x and y components of the resultant vector. Finally, we can use the Pythagorean theorem and inverse tangent to find the magnitude and direction of the resultant vector.
The x component of the resultant vector is 81.4 * cos(34.6°) + 34.4 * sin(12.5°).
The y component of the resultant vector is -81.4 * sin(34.6°) + 34.4 * cos(12.5°).
The magnitude of the resultant vector is sqrt((x component)^2 + (y component)^2).
The direction of the resultant vector is arctan((y component) / (x component)).