First of all, sorry for my poor English and thanks for your time.
I’m having problems to understand the intuition behind the dot product.
I know how to calculate the dot product with the algebraically and geometrically definitions, and I understand why are the same thanks to the Law of the Cosines:
Algebraically: $u \cdot v = u_xv_x + u_yv_y$
Geometrically: $u \cdot v = \|u\| \|v\| \cos \theta$
But when I read some definitions like: “The dot product tells you what amount of one vector goes in the direction of another” I get confused.
I’m barely understands the physics intuition of an object pull with some force vector in some distance vector with different directions and that the result of the dot product is the amount of work.
But I don't quite understand the geometrical intuition.
The result of the dot product is the length of the projected vector ($\|A\| \cos \theta$ ) multiplied with the length of the vector B($\|B\|$) .
When you calculates the dot product with at least one unit vector the result makes sense because is the length of the projected vector (because it has been multiplied by the length of the unit vector that is 1), something that you can see and identified in the space.
But when you calculate the dot product with two NO normalized vectors the result scalar it is something much bigger than any vector length and I don't understand what it represents.
Can you help me to understand the dot product intuition in a geometrically way?
I believe you are asking for too much at once. For one thing, the dot product DOES match with something - it matches with the dot product! Sometimes this has physical meaning. For example, the amount of work done (in the sense of physics) is equal to the dot product of the force and the distance through which it works. So there is at least one class of interpretations, those from physics, in which "the dot product of these two vectors is this important quantity".
Now, I believe you are looking for a purely geometry interpretation. Here, too, you are asking for too much. To see why, let's look at a much more important dot product, $||\vec{u}||\cdot||\vec{u}||$. Even if the vector is a unit vector, this dot product has an extremely important mathematical meaning - it is the square of the norm of the vector. However, this is putting the cart before the horse (that is, this definition is backwards). In many instances, this dot product is what defines the norm in the first place. Therefore, the interpretation is "the scalar that defines the norm on the particular vector space we are looking at".
To that end, the dot product of two dissimilar vectors is the "angle scaled norm product", if you'd like that geometric interpretation. It is the product of the norms reduced by the cosine of the angles between the vectors. In this sense, it might be construed as a "measure of parallel-ness" - the closer this product is to the product of the norms themselves, the closer the vectors are to parallel.