B Newton's Third Law textbook -- question about the sign of a force

Newton's Third Law: For every force acting, there is an equal and opposite force acting. Equal in magnitude, opposite in direction.
Since the "opposite in direction" is often taken as a given, other descriptions generally concentrate on the magnitude, and so FAB = FBA.

 

If there is a diagram showing the opposite arrows, they are often meant to indicate the convention for positive direction of each force individually. So the later math doesn't use vectors in one coordinate system, but factors for the indicated unit vectors. A negative result then indicates that the force is opposite to its arrow in the diagram.

 

The thing is, for a given object (free body diagram) that force only applies once. If you stick with the data you've been given in any question or practical situation, the sign of any force is either given or is calculable. You can give an 'unknown' an arbitrary sign and the calculation will produce the correct sign eventually.