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.

Any ideas?

  • 2
    $\begingroup$ Your function is dependent on your choice of orthonormal basis. One can rotate the two vectors and get a different answer. $\endgroup$ – Ishan Banerjee Dec 31 '14 at 12:36
  • $\begingroup$ That's a really interesting observation. I suppose that means hints there are not actually any geometrical interpretations? $\endgroup$ – Thomas Ahle Jan 1 '15 at 17:13

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