Product of inner products Is product of innerproduct again a inner product of two vectors? 
For example - Is $ (< u,v >)(< x,y >) = < m,n > $? And if yes is m and n unique and how do we calculate those?  

Yes, it's possible to find such vector, in your example the answer could by (for example):
$$m=<u,v>x, n=y$$
or 
$$m=<x,y>u, n=v$$ 
So it isn't unique.
