The following lame justification is often used for introducing homogeneous coordinates. Rotations and translations don't play together well. While this is true, there are actually much more siginifcant reasons for using homogeneous coordinates.
Let's take a look at why rotations and translations don't mix in a very satisfying way. Consider the following example where a point is first rotated, the translated, and then rotated again.
You can see from this example that rotation is bascially a multiplication process (albeit a matrix multiplication), whereas translation is basically an addition process.
There is a clever way to remedy this problem. This requies the use of Homogeneous coordinates.... The fact is what we really need to do is move our object into a different type of space where the origin is not a special point. Remember how all vectors were perpendicular to the origin, that's just one of the thing that makes it special. This new space is called projective space, and the reason that it has no special origin is that we exclude the origin from this space.
The origin had another special attribute, it was the starting point of all vectors. Thu, we only needed 3 basis vectors to establish a coordinate system,since we could assume that each vector originated form our "special origin". In projective space however, we'll need to specify a fourth point in order to set up a basis, since no point is any more special than any other.
Lets consider a more formal definition of projective space.
Thus projective space adds a new coordinate to each vector, but not a dimension. Each point in projective space has many possible representitives all of which vary by a non-zero scale factor. I know you're asking how is this going to help us.
We can move a point into projective space by adding a dimension like so.
now because we have freed ourselves from the oppression of an origin we can now write Rigid-body transforms as simple matrix multiplications.
