Define an x-y coordinate system such that
The positive x-axis = the eastern direction, with unit vector [tex]\hat{i}[/tex].
The positive y-axis = the northern direction, with unit vector [tex]\hat{j}[/tex].
The airplane flies at 340 km/h at 12° east of north. Its velocity vector is
[tex]\vec{v}_{1} = 340(sin(15^{o})\hat{i} + cos(15^{o})\hat{j} ) = 88\hat{i} + 328.4\hat{j}[/tex]
The wind blows at 40 km/h in the direction 34° south of east. Its velocity vector is
[tex]\vec{v}_{2} =40(cos(34^{o})\hat{i} - sin(24^{o})]\hat{j}) = 33.1615\hat{i} -22.3677\hat{j})[/tex]
The plane's actual velocity is the vector sum of the two velocities. It is
[tex]\vec{v}=\vec{v}_{1}+\vec{v}_{2} = 121.1615\hat{i}+306.0473\hat{j}[/tex]
The magnitude of the actual velocity is
v = √(121.1615² + 306.0473²) = 329.158 km/h
The angle that the velocity makes north of east is
tan⁻¹ (306.04733/121.1615) = 21.6°
Answer:
The actual velocity is 329.2 km/h at 21.6° north of east.
