I want to know how the dot product can determine whether two vectors are similar? I know that the formula $$\cos(\theta) = \frac{u \cdot v }{ ||u||\,||v||}$$ means something, but don't know what.
[Math] how does the dot product determine similarity
vectors
Related Solutions
The dot product tells you what amount of one vector goes in the direction of another. For instance, if you pulled a box 10 meters at an inclined angle, there is a horizontal component and a vertical component to your force vector. So the dot product in this case would give you the amount of force going in the direction of the displacement, or in the direction that the box moved. This is important because work is defined to be force multiplied by displacement, but the force here is defined to be the force in the direction of the displacement.
Write $a \cdot b = ( |a|) ( |b| \cos \theta)$
Then consider 
The projection of B onto A is of length $|b| \cos \theta$, it is the portion of the green line up to the dotted line perpendicular to it (or I suppose beyond it). Then the dot product is the just magnitude of these two vectors, $a$ and $b\cos \theta$, multiplied.
Note: I have interchanged $a$ and $A$, $b$ and $B$ due to the diagram and your question.
Have a look in the preview of this book, Section 1.2 has a good explanation of the dot product:
Best Answer
The dot product of two vectors $\mathbf{u}$ and $\mathbf{v}$ is defined as
$$\mathbf{u}\cdot\mathbf{v} = |\mathbf{u}|\,|\mathbf{v}|\cos \theta$$
It's perhaps easiest to visualize its use as a similarity measure when $|\mathbf{v}|=1$, as in the diagram below, where $\cos\theta = \mathbf{u}\cdot\mathbf{v}\,/\,|\mathbf{u}|\,|\mathbf{v}| = \mathbf{u}\cdot\mathbf{v}\,/\,|\mathbf{u}|$.
Here you can see that when $\theta=0$ and $\cos\theta=1$, i.e. the vectors are colinear, the dot product is the product of the magnitudes of the vectors. When $\theta$ is a right angle, and $\cos\theta=0$, i.e. the vectors are orthogonal, the dot product is $0$. In general $\cos\theta$ tells you the similarity in terms of the direction of the vectors (it is $-1$ when they point in opposite directions). This holds as the number of dimensions is increased, and $\cos\theta$ has important uses as a similarity measure in multi-dimensional space.