Q:

How does the dot product "remove" unit vectors?

From my understanding from university physics from Pearson (Page ~20):
Dot product multiplies "like" unit vector terms and cross product multiplies "unlike"  unit vector terms. So, why does the cross product retain its unit vector (ñ), but dot product does not?
Example:
$A\cdot B = AB\cos(\theta) =$ magnitude of $A$ * The component of $B$ that is parallel to $A$. 
So shouldn't the unit vector be going in the direction of $A$? Do mathematicians drop unit vectors when stuff only goes in one direction and then call it a scalar?

A:

The cross product is a rather special beast, only for $3$-dimensional space.  A more general version of it is the exterior product (wedge product), which maps into a different vector space (the exterior square), which happens in the case of dimension $3$ to be identifiable with the original space.
Dot products, on the other hand, can be defined in any finite number of dimensions.

A:

Hint: Suppose dot products were vectors and not scalars. What direction should they have? 

