Explanation:
Assuming that all forces extend from the origin point, with [tex]F_1[/tex] and [tex]F_2[/tex] lying in the xy plane, so [tex]F_3[/tex] is along the z axis. So, we have:
[tex]\vec{F_1}=F_1\hat{i}+0\hat{j}+0\hat{k}\\\vec{F_2}=F_2cos\theta\hat{i}+F_2sin\theta\hat{j}+0\hat{k}\\\vec{F_3}=0\hat{i}+0\hat{j}+F_3\hat{k}[/tex]
The net force is:
[tex]\vec{F}=\vec{F_1}+\vec{F_2}+\vec{F_3}\\\vec{F}=(F_1+F_2cos\theta)\hat{i}+F_2sin\theta\hat{j}+F_3\hat{k}[/tex]
The force ([tex]F_4[/tex]) that would exactly counterbalance these three forces will be opposite in direction and equal in magnitude to the net force:
[tex]\vec{F_4}=-(F_1+F_2cos\theta)\hat{i}-F_2sin\theta\hat{j}-F_3\hat{k}\\F_4=\sqrt{(-(F_1+F_2cos\theta))^2+(-F_2sin\theta)^2+(-F_3)^2}[/tex]
