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The work done in exerting a force F on a body over a displacement d is the scalar product (or "dot product") of the F and d, that is:

W = F.d = |Fd|cos(θ),

where θ is the internal angle between the directions of F and d, and |F| and |d| are the (positive) magnitudes of the vectors F and d respectively (see diagram below).

........d

......^

..../

../ θ

/---------------> F

Note that this formula for the work automatically gives the right sign for W, i.e. W > 0 when positive work is done on the body by the force F.

This is just the product of the force's magnitude with the component of displacement in the direction of the force. Equivalently, it is the product of d's magnitude with the component of F in the direction of d.

Obviously, if either F or d is zero then so is W, so in what follows assume that F and d are both non-zero. Then we have three cases to consider:

(1) If θ is less than a right angle (so the displacement has a positive component in the direction of the force), W is positive.

(2) If θ is equal to a right angle (so there is no displacement along the direction of the force), W is zero. Uniform circular motion provides an example of this case.

(3) If θ is greater than a right angle (so the displacement has a positive component opposite to the direction of the force), W is negative.

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