When will the equality involving inner product of averages of vectors holds

Question

Let $\left\lbrace x_{i} \right\rbrace _{i=1}^{k}$ and $\left\lbrace y_{i} \right\rbrace _{i=1}^{k}$ be nonzeros sequences in real Hilbert space. Define \begin{equation} \widehat{x} := \dfrac{1}{k} \sum\limits_{i=1}^{k} x_{i} \qquad \textrm{ and } \qquad \widehat{y} := \dfrac{1}{k} \sum\limits_{i=1}^{k} y_{i} . \end{equation} I need to determine in which case \begin{equation} \dfrac{1}{k} \sum\limits_{i=1}^{k} \left\langle x_{i} - \widehat{x} , y_{i} - \widehat{y} \right\rangle = \dfrac{1}{k} \sum\limits_{i=1}^{k} \left\langle x_{i} - \widehat{x} , y_{i} \right\rangle , \end{equation} or in other words, when will \begin{equation} \dfrac{1}{k} \sum\limits_{i=1}^{k} \left\langle x_{i} - \widehat{x} , \widehat{y} \right\rangle = 0 . \end{equation} At first, I thought that the above relation always holds. But on a second thought, I do not think this would be the case.

Answer 1

$$\sum\limits_{i=1}^{k} \left\langle x_{i} - \widehat{x} , \widehat{y} \right\rangle=\sum\limits_{i=1}^{k}\left( \left\langle x_{i} , \widehat{y} \right\rangle - \left\langle \widehat{x},\widehat{y} \right\rangle \right) =\sum\limits_{i=1}^{k} \left\langle x_{i} , \widehat{y} \right\rangle -\sum\limits_{i=1}^{k} \left\langle \widehat{x},\widehat{y} \right\rangle $$ and both terms are equal to to $k\left\langle \widehat{x},\widehat{y} \right\rangle$.