Apparently $\vec E \cdot \vec n = ||\vec E||$ . I can't see why this is through the math...the only thing we know about $\vec n$ is that its magnitude is $1$.
I realize we could use the formula $\vec E \cdot \vec n =||\vec E||||\vec n||\cos \theta$ here, but I'm reaching a contradiction through the other definition of dot product based on components:
If for example, $\vec n = \hat i +0\hat j$ and $\vec E = 3\hat i + 3\hat j$, then $\vec E \cdot \vec n = 3 \neq ||\vec E|| $
The dot product $\vec E \cdot \vec n$ is not in general equal to $||\vec E||$, the length of $\vec E$.
Remember, the dot product of two vectors $\vec A$ and $\vec B$ represents $$||\vec A|| \times ||\text{proj}_{\vec A}{\vec B}|| $$ where $\text{proj}_{\vec A}{\vec B}$ is the projection of $\vec B$ onto $\vec A$.
The above expression equals $$||\vec B|| \times ||\text{proj}_{\vec B}{\vec A}|| $$ which equals $$(||\vec A||)(||\vec B||)\cos(\theta)$$ and where $\theta$ is the angle between $\vec A$ and $\vec B$.
What this means in plain English is that to get the dot product of two vectors $\vec A$ and $\vec B$, you project one onto the other (say, $A$ onto $B$, which we write as $\text{proj}_{\vec B}{\vec A}$) , then multiply the length of the projection by the length of the other remaining vector (in this case, $\vec B$).
Now we can apply this to your specific problem. Looking at the last equation, we see that if $\vec n$ is a unit vector (that is, $||\vec n||=1$), then $$\vec E \cdot \vec n = (||\vec E||)(||\vec n||)\cos(\theta) = (||\vec E||)\cos(\theta).$$ This is equal to $||\vec E||$ precisely when $\cos(\theta)=1$, which is true precisely if $\theta$ is a multiple of $2\pi$, that is, when $\vec E$ and $\vec n$ point in the same direction.
On the other hand, if $\theta$ is not a multiple of $2\pi$, then $\cos(\theta) \ne 1$, so then $(||\vec E||)\cos(\theta) \ne ||\vec E||$.
Hope this helps, feel free to comment if you'd like me to elaborate.