I've reduced a problem I'm working on to a rather interesting relationship between three vectors. The result looks quite a bit like an inner product on three vectors, and I'm having trouble figuring out what to make of it:
In a finite-dimensional complex vector space $V$, and vectors $x,y \in V$ and fixed positive real scalars $c_1, \ldots, c_n$, and an orthonormal basis $\mathcal{B} = \{e_1,\ldots,e_n\}$ such that $$ x = x_{1}e_1 + \ldots + x_ne_n, \qquad y= y_1e_1 + \ldots + y_n e_n $$ I need to understand the conditions (preferably as much in terms of $x$ and $y$ as possible) under which $$ \frac{\bar{x}_1 y_1}{c_1} + \ldots + \frac{\bar{x}_n y_n}{c_n} = 0 $$
For some context I arrived at this by a Hadamard "division" of signals respresented by $x$ and $y$, by $(\sqrt{c_1},\ldots , \sqrt{c_{n}})$ and then taking their inner product, but I strongly get the sense that this second form is of some importance. It's essentially an orthogonality condition of sorts between $x,y$ and the vector $(\frac{1}{c_1},\ldots , \frac{1}{c_n})$. Or else it can be viewed like a sequilinear form of sorts, with $C$ being the diagonal matrix with entries $\frac{1}{c_i}$ and then $$ x^{*}Cy = 0. $$ I suspect this might be something that would crop up in Clifford algebra or some other kind of alien technology. But I'd be interested in any insight that anyone's got. If possible, I would love to have equivalent conditions of some sort regarding these vectors, or their coefficients in $\mathcal{B}$, or even their relative magnitudes, perhaps.