Answer
(a) $\frac{55}{2}i+\frac{55\sqrt 3}{2}j$
(b) $\frac{765\sqrt 2}{2}i+\frac{765\sqrt 2}{2}j$
(c) $568.4i+588.6j$
(d) $818.2mi/h$, N $45^{\circ}$ E
Work Step by Step
(a) The wind is blowing 55mi/h N $30^{\circ}$ E, so the angle $\theta=60^{\circ}$ from the x-axis, giving a vector of
$\vec {v_w} =55cos60^{\circ} i + 55sin60^{\circ} j=\frac{55}{2}i+\frac{55\sqrt 3}{2}j$
(b) The jet is flying 765mi/h N $45^{\circ}$ E, so the angle $\theta=45^{\circ}$ from the x-axis, giving a vector of
$\vec {v_j} =765cos45^{\circ} i + 765sin45^{\circ} j=\frac{765\sqrt 2}{2}i+\frac{765\sqrt 2}{2}j$
(c) The true velocity of the jet is the sum of the above vectors as $\vec {v_t} =\vec {v_j} +\vec {v_w} \approx568.4i+588.6j$
(d) The true speed is the modulus of the velocity: $|\vec {v_t} |=\sqrt {(568.4)^2+(588.6)^2}\approx818.2mi/h$,
and direction of the jet with respect to the x-axis is $\theta=tan^{-1}\frac{588.6}{568.4}\approx45^{\circ}$ or N $45^{\circ}$ E
