What Good Are Dot Products?

The dot product formula is easy to compute and gives you information about the angle between vectors.
Many times you don't need an exact angle - it's often enough to know whether or not two vectors are perpendicular . (How could you tell this from the dot product?)
The dot product formula has to do with two vectors. In general two vectors determine a plane. The formula extends directly to vectors in three-dimensional space (and even higher!).
Physicists think (happily) that the dot product tells you about work. If you have a constant force applied directly in the direction an object is moved, then the amount of work = (force)(distance).