Does the dot product angle formula work for $\Bbb{R}^n$?

Question

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 $\Bbb{R}^n$?

Answer 1

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.

Answer 2

Yes. It does. The inner product is defined in n space in linear algebra. https://www.princeton.edu/~achaney/tmve/wiki100k/docs/Euclidean_space.html. You should also read the wikipedia article on inner product.