Answer
a) $F_1\lt F_2=F_4\lt F_3$
b) $\vec v_1$
Work Step by Step
(a) We know that the magnetic force for $\vec v_2$ and $\vec v_4$ will be zero because the angle between the velocity vector and the magnetic field vector is $0^{\circ}$ and $180^{\circ}$. For $\vec v_1$, the direction of the magnetic force is outward, perpendicular to the plane of the paper and for $\vec v_3$, the magnetic force is downward along the direction of the electric field. The net force $\vec v_3$ will be greater than the net force for $\vec v_1$ because the electric and magnetic force are in the same direction for $\vec v_3$. Thus, the required order of increasing magnitude is $F_1\lt F_2=F_4\lt F_3$
(b) We know that the electric force and magnetic force are opposite to each other only in $\vec v_1$ in the given case. Thus, they can add up to zero if their magnitude becomes the same. For all the other cases, the magnetic force is either zero or in the direction of the electric force.
