Say $x$ and $y$ are two $L_2$ unit vectors of size $n$. In that case the inner product:
Is the cosine of the angle between them.
For an application I was originally interested in the angle like this, but I have only been able to achieve the squared inner product:
And now I wonder if this has any interesting geometrical interpretations? I suppose it can't be too closely related to the angle between the vectors, given the value can only be in the interval $[0,1]$.
I suppose this is also similar to what we have in the Cauchy Schartz inequality, after some rewriting, but I'm not sure what the geometric intepretation is of this.