Understanding multiple linear regression residuals I'm a bit confused whether residual vectors in OLS are orthogonal to every vector in X.
The problem I have is: Which of the following are true statements for our multiple linear regression with datapoints as rows of design matrix X, label vector Y, and using the model Ŷ = X̂.
I can understand that the residual vector e = Y - Ŷ is orthogonal to Y. And to me, it would make sense for the residual vector e = Y - Ŷ to also be orthogonal to some cases of vector in X. But what about every?
Any help would be appreciated.
I reference this diagram:


Assuming $X$ is full rank, you can see it in terms of the hat matrix $H = X(X^\top X)^{-1}X^\top$. This matrix is an orthogonal projection matrix onto the column space of $X$.
The estimate of $y$ is $\hat{y}=X\hat{\theta}=Hy$. The residual is $e = y - \hat{y} = (I-H)y$. The matrix $(I-H)$ is an orthogonal projection matrix onto the orthogonal complement of the column space of $X$, so the residual vector $e$ is orthogonal to every vector in the column space of $X$.
