# Geometric interpretation of $x_1^2y_1^2+x_2^2y_2^2+x_3^2y_3^2+\dots$

Say $x$ and $y$ are two $L_2$ unit vectors of size $n$. In that case the inner product:

$$x_1y_1+x_2y_2+x_3y_3+\dots+x_ny_n$$

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:

$$x_1^2y_1^2+x_2^2y_2^2+x_3^2y_3^2+\dots+x_n^2y_n^2$$

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?

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