(More) Rigorous Proof of the Formula for the Angle Between Two Vectors I looked into half a dozen proofs for the angle between vectors formula involving cosine.
$$\cos \theta = \frac{\vec v \cdot \vec w}{\|\vec v\|\|\vec w\|}$$
They all start like this:
An $n$-dimensional triangle is constructed and projected onto $\mathbb{R}^2$ somehow (<-- somehow!):

How this projection happens is not really important though, because adding $\vec v$ and $\vec w - \vec v$ lands you at $\vec w$, which demonstrates that these vectors form 'a closed path'... lacking proper terminology here.
Then some major hand waving happens, in which the vectors are replaced by their magnitudes and it is stated that thus the law of consines can be applied to this triangle, because the vectors have ben replaced by their lengths.

But... that is not rigorous. Or I at least it makes it not explicit why this step is valid, i.e. for instance what the property is called that is proved to hold in this case.
I mean, who says that just because the $n$-D vectors can be arranged to form a closed path (demonstrably), that we can just willy nilly assume that we can apply the law of consines to a triangle constructed from the magnitudes of the vectors?
Can you provide me with the reason why this step is allowed and ideally a proof as well?
Examples for proofs that do this magic step (Yes, I don't like reading):
https://www.youtube.com/watch?v=bbBGgHDhmVg
https://www.youtube.com/watch?v=adOtjgAeIbE
https://www.youtube.com/watch?v=5AWob_z74Ks
 A: It seems that you define the scalar product $x\cdot y$ in ${\mathbb R}^n$  by
$$x\cdot y=\sum_{k=1}^n x_k\, y_k\ ,$$
whereby $(x_k)_{1\leq k\leq n}$ are the coordinates of $x$ with respect to the standard basis. This basis  is orthonormal. It is a standard fact that formula $(1)$ is valid with respect to  any orthonormal basis $(e_k)_{1\leq k\leq n}$ of the vector space ${\mathbb R}^n$. 
Now to angles between vectors. The angle $\theta$ between two nonzero vectors $u$ and $v$ is defined to be the shorter of the two arcs on the unit circle connecting the tips of the unit vectors $u/|u|$ and $v/|v|$. It is stated that this angle $\theta$ between two vectors $u$, $v\in{\mathbb R}^n$ is characterized by $$\cos\theta={u\cdot v\over |u|\>|v|}\ .$$
We therefore have to prove the claim that
$$\cos\theta=u\cdot v\qquad\bigl(|u|=|v|=1\bigr)\ .$$
To this end choose an orthonormal basis $(e_k)_{1\leq k\leq n}$ with $e_1:=u$ and $v\in\langle e_1,e_2\rangle$. Then all coordinates  $u_k$ and $v_k$ with $k>2$ are zero, and we have $$u\cdot v=1\>v_1+0\>v_2=v_1=\cos\theta\ ,$$ because the cosine of the angle $\theta$ enclosed by $u$ and $v$ is by definition the first coordinate  of $v$ when $u=e_1$ and $v$ is another point of $S^1$ in the $(x_1,x_2)$-plane. 
