The Pearson's product–moment correlation coefficient

Question

The Pearson's product–moment correlation coefficient is $ r_{xy}=\dfrac{\sum_{i=i}^n(x_i-\overline{x})(y_i-\overline{y})}{\sqrt{\sum_{i=i}^n(x_i-\overline{x})^2}\sqrt{\sum_{i=i}^n(y_i-\overline{y})^2}}, $ where

$n$ is sample size.

$x_{i},y_{i}$ are the individual sample points indexed with $i$.

$\overline{x}=\dfrac{1}{n}\sum_{i=1}^nx_i$ (the sample mean); and analogously for ${\bar {y}}$.

Let $\mathbf{u}=\begin{bmatrix} x_1-\overline{x}\\ \vdots\\ x_n-\overline{x}\\ \end{bmatrix}$, $\mathbf{v}=\begin{bmatrix} y_1-\overline{y}\\ \vdots\\ y_n-\overline{y}\\ \end{bmatrix}$, and $\theta$ be the angle between $\mathbf{u}$ and $\mathbf{v}$.

Clearly, $r_{xy}=\dfrac{\mathbf{u}\cdot\mathbf{v}}{\lVert \mathbf{u}\rVert\lVert \mathbf{v}\rVert}=\cos \theta$.

My questions:

1. Is there any geometric meaning/deeper meaning of the $\theta$? Why is $r_{xy}$ is equal to $\cos \theta$?

2. Can we see the $\theta$ in the graph of $y$ vs $x$?