In order for the object to continue with constant velocity (no acceleration), the forces on it must be balanced. That means they all have to add up to zero.
-- The 3rd force has to cancel 80N north, so part of it has to be 80N south.
-- It also has to cancel 60N west, so another part of it has to be 60N east.
The combined magnitude of these two 'components' is
Mag = √(80² + 60²)
Mag = √(6,400 + 3,600)
Mag = √(10,000)
Magnitude = 100 Newtons
For the direction of the new force . . .
We know it'll be pushing partly toward South and partly toward East. So when you put these together, it's pushing somewhere in the neighborhood of Southeast.
If you draw yourself a picture of the two components, and then draw the resultant (the combination of both), you'll have a right triangle. You'll see that the components are the legs of the triangle, and the angle 'below' east has a tangent that's (the 80) divided by (the 60) ... which is 1.333 .
Now you go to your calculator and find the angle that has a tangent of 1.333 . That's 53.13° .
So the new third force, with a magnitude of 100 Newtons, has to point in the direction of 53.13° south of east, in order to exactly cancel out the other two forces. There's no other way to do it with just one single new force.
