In inner product spaces, angle is defined to be the only $0 \leq \theta \leq \pi$ satisfies: $$\cos\theta=\frac{\left<v,u\right>}{\left\|v\right\|\left\|u\right\|}$$ where $u,v\in V$ - an inner product space.
I wondered why this definition, that is abstract in most inner product spaces, converges with the geometric definition of an angle in $ \mathbb{R}^{3}$ with the standard inner product. Does someone have an explnation?
Thanks!
If you apply $\left<v,u\right>$, you basically project the vector $\vec v$ in the direction of $\vec u$, which is just the same as the definition of $\cos\theta$ .