Answer
a) and c) See the figure below.
b) $359.5\;{\rm m}, 59.4^\circ $
Work Step by Step
a)
We drew an accurate graphical representation of Bob’s motion as you see in the figure below.
b)
We need to find the vector that made Bob reach home again.
Thus,
$$\vec D=-\left(\vec A+\vec B+\vec C\right)$$
$$\vec D=-\left( 0+ 400\cos[180^\circ+45^\circ]+ 200\cos60^\circ \right) \hat i-\left( -200+ 400\sin[180^\circ+45^\circ]+ 200\sin60^\circ \right) \hat j$$
$$\vec D=-(182.8\;{\rm m})\hat i-(309.6\;{\rm m})\hat j$$
The magnitude of $\vec D$ is given by applying the Pythagorean theorem.
$$|\vec D|=\sqrt{D_x^2+D_y^2}=\sqrt{(-182.8)^2+(-309.6)^2}=\color{red}{\bf 359.5}\;\rm m$$
and its direction is given by
$$\tan\alpha_D=\dfrac{D_y}{D_x}$$
Thus,
$$\alpha_D=\tan^{-1}\left[\dfrac{D_y}{D_x}\right]=\tan^{-1}\left[\dfrac{-309.6}{-182.8}\right] $$
Thus,
$$\alpha_D=\color{red}{\bf 59.4^\circ}$$
As we can see, in the second figure below, the two results are so close.
