What we have learned about energy until now can be summarized in two equations:
where the work performed takes the form
This is the most general form for work done.
Sometimes there are simplifying
circumstances which also simplify the expression for W. If the force
does not depend on position, 
then is can go outside the
integration:
where we have defined
as the total displacement vector in the motion from 
to
.
Note that the
``
'' in the expressions above
denotes a scalar product between two vectors which is defined as
where 
is the angle between the vectors. In words what this means
in the expression for work is that only the projection of the force vector on the
direction of displacement contributes to work done. The perpendicular component
while it will affect the motion it will not change the kinetic energy of the object
in motion. Recalling the our discussion of circular motion the component of force
perpendicular to displacement generally just alters the direction of motion, not the
speed.
Further simplifications result if the motion takes place in one dimension only. Then
th scalar product simplifies to a regular product and
and in the very simplest case of a constant force in one dimension we end up
with
Work performed by gravity close to earth involves the constant
force 
so we can use Eq. 4 to obtain
the expression:
Note that this expression holds for any crazy displacement we might choose.