Rotations: transformations that preserve length of and angle between, at least, two vectors

Wikipedia has a nice article on the SO(3) group where they start by stating that:
Rotation Group SO(3)

"length-preserving transformation in R³ preserves the dot product, and thus the angle between vectors"

They then continue by stating that:
$$
\vec{u}\cdot\vec{v} =
\frac{1}{2}\left(
  \left\|\vec{u} + \vec{v}\right\|^{2} - \left\|\vec{u}\right\|^{2} - \left\|\vec{v}\right\|^{2}
\right)
$$
However, the reader may not be aware of the fact that:
$$
\vec{u}\cdot\vec{v} = \left\|\vec{u}\right\|\left\|\vec{v}\right\|\cos\left(\theta\right)
$$
or

$$
 \cos\left(\theta\right)=\frac{\vec{u}\cdot\vec{v}}{\left\|\vec{u}\right\|\left\|\vec{v}\right\|}
$$

where one can explicitly see that, by definition of the dot product, if the coordinate system of \(\vec{u}\) and \(\vec{v}\) undergoes any rotation,  where the dot product is conserved, and we know that the lengths of \(\vec{u}\) and \(\vec{v}\) are conserved, then the angle \(\theta\), that is the angle \(\angle uOv\), can not change and is, therefore, also conserved.

(Really, when I saw that Wikipedia ignored readers that didn't know the definition of the dot (inner) product, I felt this was just a good excuse to play with latex in blogger. Though it left me wanting to be able to post graphs & vector diagrams in my blog, a la LaTeX.)