A standard explanation of tidal forces on the earth shows the earth with three force vectors pointing toward the moon. The vector on the side of the earth nearest the moon, Fn, is the largest and the vector on the side of the earth farthest from the moon, Ff, is the smallest. The vector at the center of the earth, call it Fc, is in-between the other two in magnitude.
Then what is done is to take the differences Fn - Fc and Ff - Fc. Taking the difference of force vectors acting on the same object gives the net force on that object. What does it mean to take the difference of force vectors acting on different objects?
Note: I am not after an explanation/discussion of tidal forces per se, only comment on the above approach used in many places.