MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Physics
Physics 8.01
W01D2-1 Vector Addition of Forces Solution
Three forces are acting on an object that is at rest. Two of those forces are given by the

vector expressions F1 = 70 N î + 20 N ĵ and
as shown in the figure
above. Find the direction and magnitude of the third force.
Solution: The sum of the forces on the object is zero because the object is at rest,
therefore
   
0 = F1 + F2 + F3 .

We can now determine the vector F3 :

 
F3 = −(F1 + F2 ) = −(F1x + F2 x ) î − (F1y + F2 y ) ĵ
= −(70 N + − 30 N) î − (20 N + 40 N) ĵ
= −(40 N) î − (60 N) ĵ

The magnitude is F3 = (−40 N)2 + (−60 N)2 = 72 N . The angle the force makes with
the positive x-axis is determined by
⎡ −60 N ⎤
θ = tan −1 ⎢
⎥ = 56.3° or 236.3° .
⎣ −40 N ⎦

There are two solutions. The angle θ = 236.3° corresponds to the vector F3 as shown in

the figure below. The angle θ = 56.3° corresponds to the vector −F3 .