Three collinear forces, F1=45N west, F2=63N east and an unknown force F3 are applied to an object. The resultant force of the three forces is 12N west. Show by means of a tail to head method of vector addition how the magnitude of F3 could be found?

Take east to be the positive direction. Then the resultant force from adding F₁ and F₂ is

F₁ + F₂ = (-45 N) + 63 N = 18 N

which is positive, so it's directed east.

To this we add a third force F₃ such that the resultant is 12 N pointing west, making it negative, so that

18 N + F₃ = -12 N

F₃ = -30 N

So F₃ has a magnitude of 30 N and points west.

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