If there is an external force on a particle causing it to move in a certain direction by a displacement s, then there must be a component of that force acting in that direction and doing the work.

This “work” is a mechanical quantity and is defined to be the product of that force times the displacement it moves:

W = **F**.**s**

Here, we have taken the *dot product* of the two vector quantities **F** and **s**. The dot product has the property that it takes the component of F which is in the direction of d and multiplies them to get a scalar quantity, W.

If the force F is applied at an angle θ, the component along the displacement is F cos θ

and hence work becomes

W = F s cos θ

In the case of a varying force, net work done during a small displacement d**s** can be written as:

dW = **F**.d**s**

and the total work done along a path from initial point to the final point can be written as:

W =∫ dW = ∫ **F**.d**s**