**Pre Reqs:** [What is a Force](http://blog.dotphys.net/2008/09/basics-what-is-a-force/)

[Previously, I talked about the momentum principle](http://blog.dotphys.net/2008/10/basics-forces-and-the-momentum-principle/). Very useful and very fundamental idea. The other big (and useful) idea in introductory physics is the work-energy theorem. Really, with work-energy and momentum principle, you will be like a Jedi with a lightsaber and The Force – extremely powerful.

Well, what is work? What is energy? How are they related? In [another post, I talked about energy.](http://blog.dotphys.net/2008/10/what-is-energy/) I think it is interesting to look at how most textbooks define energy:

*Energy is the ability to do work*

This is really a stupid definition. Kind of circular logic, if you ask me. In the post I mentioned earlier, I claim there are two kinds of energy, particle energy and field energy. At low speeds (not near the speed of light), particle energy can be written as:

![Screenshot 53](http://blog.dotphys.net/wp-content/uploads/2008/10/screenshot-53.jpg)

Where *m* is the mass of the particle, *c* is the speed of light. So, if you just look at a particle, that is it for the energy. Now, what about the “work” portion? Work is defined as:

![Screenshot 54](http://blog.dotphys.net/wp-content/uploads/2008/10/screenshot-54.jpg)

Where *F* is the net force on the particle, ?r is the vector displacement of the particle. The “dot” in between F and ?r represents the “dot product” operation between vectors (also known as the scalar product). In a [previous post](http://blog.dotphys.net/2008/09/basics-vectors-and-vector-addition/) I showed that you could multiply a scalar quantity by a vector quantity. Here I need to do “something” with two vectors. You can’t multiply two vectors in the same sense that you multiply scalars. A general definition of the dot product for two vectors:

![Screenshot 55](http://blog.dotphys.net/wp-content/uploads/2008/10/screenshot-55.jpg)

That looks a little more messy than I wanted, but it can not be helped. Really, it is not that complicated. The dot product is simply the projection of one vector on the other. Let me explain in terms of work.