Consider vectors $x_1, \cdots, x_n \in \mathbb{R}^m$. Define the vector $\mu \in \mathbb{R}^m$ to be the mean of the vectors:
$$ \mu = \frac{1}{n}\sum_{i=1}^n x_i $$
Assume that $\mu = 0$, the zero vector.
Now consider some other vector $u$. Define the orthogonal project of $x_i$ onto $u$ as $v_i$, a scalar, i.e., $$ v_i = u^Tx_i $$
And define $\phi \in \mathbb{R}$ as the mean of $v_i$. Is $\phi = \frac{1}{n}\sum_{i=1}^n v_i = 0$?
The way I approached this problem seems trivial, and I'm not sure if I did this correctly. Essentially, I started with
$$ \mu = \frac{1}{n}\sum_{i=1}^n x_i = 0 \\ u^T\frac{1}{n}\sum_{i=1}^n x_i = u^T0 \\ = \frac{1}{n}\sum_{i=1}^n u^Tx_i = \phi = 0 \\ \therefore \phi = 0 $$
This seems like a trivial proof if I did this correctly, but I'm having a hard time visualizing why this is true, i.e., why is the mean of the orthogonal project of the vectors onto $u$ zero?
This result also appears to be independent of $u$.
Orthogonal projections are linear maps.
And linear maps: (1) map the zero vector to the zero vector and (2) map the mean of vectors to the mean of the mapped vectors.