Answer
(a) $2i-4j$
(b) $-2i+4j$
Work Step by Step
(a) Given $\vec {F_1} =3i-j$, $\vec {F_2} =i+2j$, $\vec {F_3} =-2i-j$, and $\vec {F_4} =-4j$, the resultant force acting at P is the sum of the forces $\vec {F} =\vec {F_1} + \vec {F_2} + \vec {F_3} +\vec {F_4} =2i-4j$
(b) For the forces to be in equilibrium, the additional force required should be $\vec {F_a} =-2i+4j$
which satisfies the equation $\vec {F} + \vec {F_a} =0$
