The angle between two vectors $\vec{v}$ and $\vec{w}\in \mathbb{R}^2$ of length $1$ is the number $\theta\in [0,2\pi)$, for which it holds that $\cos\theta=\langle \vec{v}\mid \vec{w}\rangle$.
Show using a right triangle why this is equivalent to the geometric definition of cosine (in the triangle).
Why do we need that the vectors have length $1$ ?
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From the triangle we get that $\cos\theta=\frac{\text{adjacent side}}{\text{hypotenuse}}$. How do we get from that the scalar product $\langle \vec{v}\mid \vec{w}\rangle$ ?