If $\langle \boldsymbol{x}, \boldsymbol{y} \rangle = 0$, where $\boldsymbol{x}, \boldsymbol{y}$ are vectors, then $\langle \boldsymbol{x} - \mu_\boldsymbol{x} \boldsymbol{1}, \boldsymbol{y} - \mu_\boldsymbol{y} \boldsymbol{1} \rangle = 0$? $\mu_x$ and $\mu_y$ are the means of the vector $x$ and $y$.
I proved this below $$ \langle \boldsymbol{x} - \mu_\boldsymbol{x} \boldsymbol{1}, \boldsymbol{y} - \mu_\boldsymbol{y} \boldsymbol{1} \rangle \\ = \langle \boldsymbol{x} , \boldsymbol{y} \rangle - \langle \mu_x \boldsymbol{1} , \boldsymbol{y} \rangle - \langle \mu_y \boldsymbol{1} , \boldsymbol{x} \rangle + \langle \mu_y \boldsymbol{1}, \mu_x \boldsymbol{1} \rangle \\ = 0 - \langle \mu_x \boldsymbol{1} , \boldsymbol{y} \rangle - \langle \mu_y \boldsymbol{1} , \boldsymbol{x} \rangle + \mu_x \mu_y \langle \boldsymbol{1}, \boldsymbol{1} \rangle \\ \langle \mu_y \boldsymbol{1} , \boldsymbol{x} \rangle = \left( \frac{\sum_i y_i}{N} \right) (\sum_i x_i) \\ \langle \mu_x \boldsymbol{1} , \boldsymbol{y} \rangle = \left( \frac{\sum_i x_i}{N} \right) \sum_i y_i \\ \mu_x \mu_y \langle \boldsymbol{1}, \boldsymbol{1} \rangle = 2N \left( \frac{\sum_i y_i}{N} \right) \left( \frac{\sum_i x_i}{N} \right) $$ So we can conclude the expression evaluates to zero. But I am having trouble making sense of this intuitively. Is there an intuitive explanation for why this holds?