Source: Principles of Physics
Problem Sets:
Problem
A particle moves with a constant velocity $\vec{v}$. Three forces, $\vec{F}_1$, $\vec{F}_2$, and $\vec{F}_3$, act on it. The first two forces are given by $\vec{F}_1 = F_{1x}\hat{i} + F_{1y}\hat{j} + F_{1z}\hat{k}$ and $\vec{F}_2 = F_{2x}\hat{i} + F_{2y}\hat{j} + F_{2z}\hat{k}$.
Since the particle moves at a constant velocity, its acceleration is zero, $\vec{a} = 0$. According to Newton's second law, the net force on the particle must be zero.
$$\sum \vec{F} = \vec{F}_1 + \vec{F}_2 + \vec{F}_3 = m\vec{a} = 0$$Solving for the third force $\vec{F}_3$:
$$\vec{F}_3 = -(\vec{F}_1 + \vec{F}_2)$$The sum of the first two forces is found by adding their respective components:
$$\vec{F}_1 + \vec{F}_2 = (F_{1x} + F_{2x})\hat{i} + (F_{1y} + F_{2y})\hat{j} + (F_{1z} + F_{2z})\hat{k}$$Therefore, the third force is the negative of this sum.