Answer
a)
The resultant force is $-2\mathbf{j}$.
b) The equilibrium force is $2\mathbf{j}$.
The force is at equilibrium when the net force is 0.
Work Step by Step
(a)
If the force ${{\mathbf{F}}_{\mathbf{1}}}$ in the rectangular component is $\left( -3,0 \right)$.
Then force in vector form is ${{\mathbf{F}}_{\mathbf{1}}}=-3\mathbf{i}$.
If the force ${{\mathbf{F}}_{2}}$ in the rectangular component is $\left( -1,4 \right)$.
Then force in vector form is ${{\mathbf{F}}_{2}}=-\mathbf{i}\text{+4}\mathbf{j}$.
If the force ${{\mathbf{F}}_{3}}$ in the rectangular component is $\left( 4,-2 \right)$.
Then force in vector form is ${{\mathbf{F}}_{3}}=4\mathbf{i}-2\mathbf{j}$.
If the force ${{\mathbf{F}}_{4}}$ in the rectangular component is $\left( 0,-4 \right)$.
Then force in vector form is ${{\mathbf{F}}_{4}}=-4\mathbf{j}$.
The resultant force is given by $\mathbf{F}$:
$\begin{align}
& \mathbf{F}={{\mathbf{F}}_{1}}+{{\mathbf{F}}_{2}}+{{\mathbf{F}}_{3}}+{{\mathbf{F}}_{4}} \\
& {{\mathbf{F}}_{1}}+{{\mathbf{F}}_{2}}+{{\mathbf{F}}_{3}}+{{\mathbf{F}}_{4}}=-3\mathbf{i}+\left( -\mathbf{i}\text{+4}\mathbf{j} \right)+\left( 4\mathbf{i}-2\mathbf{j} \right)+\left( -4\mathbf{j} \right) \\
& =-3\mathbf{i}-\mathbf{i}+4\mathbf{i}+\text{4}\mathbf{j}-2\mathbf{j}-4\mathbf{j} \\
& =-2\mathbf{j}
\end{align}$
(b)
Let the equilibrium force be given by $\mathbf{G}$.
Then,
$\begin{align}
& \mathbf{F}+\mathbf{G}\text{=0} \\
& \mathbf{G}=\mathbf{-F} \\
\end{align}$
Put the value of $\mathbf{F}=-2\mathbf{j}$ in above equation:
$\begin{align}
& \mathbf{G}=-\left( -2\mathbf{j} \right) \\
& =2\mathbf{j}
\end{align}$
