No correlation between explanatory variables and residuals in a linear model Suppose we have a linear model $Y=X \beta + \epsilon$ where $\epsilon \sim N(0, \sigma^2 I)$ and let $H$ be the projection matrix into the column space of $X$.
We define the residual $e$ as the projection of $Y$ into span$(X)$ (i.e. $e=(I-H)\epsilon \text{ }$).
A statement says that : by construction, $X^T e=0$ and it implies that no correlation should appear between explanatory variables and residuals.
I can convince myself that the statement is true by saying that if $X_i$ is a column of $X$, then using the fact that $E(e)=0$, we have $Cov(X_i,e)=\Bbb E(X_i. e)- \Bbb E(X_i )\Bbb E(e)=0 \implies Corr(X_i,e)=0$.
However, I don't really intuitively see why there should or not be any correlation between explanatory variables and residuals. Can someone explain the intuition behind the statement ?

Nick Cox answered OP's query in the comment:

The residuals are what you haven't explained with your explanatory variables. That's how they are defined.

Take a quick glance at what the least square does geometrically:

In the figure, $\Omega := \mathcal C(\mathbf X). $ One needs to seek $\boldsymbol\theta \in \mathcal C(\mathbf X) $ such that $\rm AB$ is minimum. From the figure, it can be noticed $\hat{\boldsymbol\theta}$ does the job. Method of least squares yields the desired $\hat{\boldsymbol\theta}$ which is the projection of $\mathbf Y$ on $\Omega.$ Again noticing the figure dictates us the following:
$$\left(\mathbf Y-\hat{\boldsymbol\theta}\right) \perp \Omega.\tag 1\label 1$$
What is $\mathbf Y-\hat{\boldsymbol\theta}? $ It is the residual vector $\mathbf e. $ So, what $\eqref 1$ is implying is that $\mathbf e$ is orthogonal to $\mathcal C(\mathbf X) $ that is, $\mathbf e$ cannot be expressed as the linear combination of the columns of $\mathbf X. $

Source of the figure:
$\rm [I]$ Linear Regression Analysis, George A. F. Seber, Alan J. Lee, John Wiley & Sons, $2003, $ Fig. $3.1, $ p. $37.$
