Why is vectors A.B expressed in the form of $a_xb_x +a_yb_y +a_zb_z$ in the dot product? I don't know how to put the subscript x and y there....but I hope that you will know what I mean. I understand the other definition of the dot product involving cos...but not this one. How is this way derived? Can someone help me make intuitive sense of this? Why must we add the xyz components?
Another question is why the dot product is a scalar quantity? If I resolve vector $A$ to a vector $B$, shouldn't I get a new magnitude in the direction of vector $B$?
 A: Well, in a general way the dot product is an operation defined between two objects that satisfies some properties (check out this page on wikipedia for example: http://en.wikipedia.org/wiki/Dot_product), for a 3D vector it turns out that if you define the dot product as: 
$$ \mathbf{A} \cdot \mathbf{B} = a_xb_x + a_yb_y + a_zb_z  $$ 
these properties are fullfilled. 
Note that $ a_xb_x + a_yb_y + a_zb_z  $ is ascalar and not a vector anymore, because you are summing up products between scalar quantities (i.e. the components of your vectors).
Now consider the dot product between the vector $ \mathbf{A}= a_x \mathbf{i} + a_y \mathbf{j} + a_z \mathbf{k} $ and the vector $  \mathbf{i} $, using the formula above, you get: 
$$ \mathbf{A} \cdot \mathbf{i} = a_x = \mid \mathbf{A} \mid cos\theta  $$ 
where $\theta$ is the angle between $  \mathbf{A} $ and $  \mathbf{i} $, i.e. $ \mathbf{A} \cdot \mathbf{i}$ gives you the projection of $ \mathbf{A} $ on $\mathbf{i} $ (try to sketch the vectors in 2D). 
So, in a general way, you have that $ \mathbf{A} \cdot \mathbf{B}  $ gives you the projection of $\mathbf{A} $ on $\mathbf{B}$, i.e $ \mid \mathbf{A} \mid \mid \mathbf{B} \mid cos\theta  $.
