Vector forces equilibrium

Question

Two forces, represented by the vectors $\vec f=2\vec i-8\vec j$ and $\vec g=5\vec i-4\vec j$ are acting on an object. Give a vector $\vec h$ representing the force that must be applied to the object if it is to remain stationary.

Answer 1

Simply note that, for Newton's first law, the resulting force $\vec r$ must be equilibrated by $\vec h$:

$$\vec r=\vec f+\vec g=2\vec i-8\vec j+5\vec i-4\vec j=7\vec i-12\vec j\implies \vec h=-\vec r=-7\vec i+12\vec j$$