Finding the missing equilibrium force from vector components

Key concept: This is a vector equilibrium problem. The missing force is found by making the total vector sum equal to zero.

Step by step

Concept being tested

When forces are in equilibrium, the vector sum of all forces is zero. That means the missing force must exactly cancel the combined effect of the other forces.

Here the known forces are

$F_1 = \begin{pmatrix} 30 \\ -85 \end{pmatrix}, \quad F_2 = \begin{pmatrix} 108 \\ 40 \end{pmatrix}$

and $F_3$ is the force needed to keep equilibrium.

Step-by-step method

For equilibrium,

$F_1 + F_2 + F_3 = \mathbf{0}.$

So

$F_3 = -(F_1 + F_2).$

First add the components:

$F_1 + F_2 = \begin{pmatrix} 30 + 108 \\ -85 + 40 \end{pmatrix} = \begin{pmatrix} 138 \\ -45 \end{pmatrix}.$

Then change the sign of each component:

$F_3 = \begin{pmatrix} -138 \\ 45 \end{pmatrix}.$

Why the sign matters

Equilibrium is not just about matching sizes; it is about matching directions too. The x-component must cancel the x-components of the other forces, and the y-component must cancel the y-components. That is why the answer is a vector with opposite components.

This is the standard inverse of vector addition: if two forces make a resultant, the equilibrium force is the negative of that resultant.

Final answer

$\boxed{F_3 = \begin{pmatrix} -138 \\ 45 \end{pmatrix}\text{ N}}$

That vector exactly balances the two given forces.

Pitfall alert

A common error is to add the vectors correctly and then forget to reverse the sign for equilibrium. Another mistake is to write the resultant vector itself as the missing force, which would make the system unbalanced rather than balanced. It also helps to keep the components aligned carefully: the top entry is the east-west component and the bottom entry is the north-south component in this notation. Swapping them changes the force completely and leads to an incorrect equilibrium condition.

Try different conditions

If the problem asked for the resultant of the two known forces instead of the missing equilibrium force, the answer would be $\begin{pmatrix} 138 \\ -45 \end{pmatrix}$ N. If one of the vectors were given in magnitude-and-direction form, you would first convert it into component form before adding. For example, a force of 50 N at an angle would need $x$- and $y$-components before the equilibrium force can be found. The same rule always applies: equilibrium means the total vector sum is zero.

Further reading

vector equilibrium, resultant force, component form