Does the dot product angle formula work for $\Bbb{R}^n$? Whenever I have seen this formula discussed 
\begin{equation}
\textbf{A} \cdot \textbf{B} = \|\textbf{A} \| \|\textbf{B} \| \cos\theta
\end{equation}
I have always seen it using vectors in $\Bbb{R}^2$. 
I was wondering if this property works if $\dim \textbf{A}= \dim \textbf{B} = n$. I feel like it wouldn't because for spherical coordinates we need more than one angle. We use $\phi$ and $\theta$. But I have no idea. Maybe I am confusing the meaning of $\theta$ in the dot product angle formula. It isn't a dimension right? 
My Question: 
Does the dot product angle formula apply to $\Bbb{R}^n$?

Yes it does. If you accept that this holds in $\mathbb R^2$, the fact that it holds in $\mathbb R^n$ follows fairly easily - notice that for any choice of two vectors $A$ and $B$, it is always possible to choose a two-dimensional subspace (i.e. a plane) containing both vectors (the span of the two vectors usually functions), and the dot product on this subspace is the same as in $\mathbb R^2$ - so we're measuring the angle between $A$ and $B$ as if they were just two dimension vectors.
