Why the angle spanned by three points of En is independent of the choice of coordinates?

Question

The Euclidean n space $(\mathbb{E}^n,d)$ is a metric space with metric $d$ for which there exists a bijective map $\mathbb{E}^n\rightarrow\mathbb{R}^n$, such that if P,Q $\in \mathbb{E}^n$ are mapped to $\mathbf{x,y}\in\mathbb{R}^n$ then $d(P,Q)=\lvert{\mathbf{y}-\mathbf{x}}\rvert$

If $P,Q,R\in\mathbb{R}^n$ are mapped to $\mathbf{x},\mathbf{y},\mathbf{z}\in\mathbb{R}^n$, then the angle spanned by P,Q,R is equal to $\angle\mathbf{xyz}$ and we can evaluate it using

$\cos(\angle\mathbf{xyz})=\frac{(\mathbf{x}-\mathbf{y})\cdot(\mathbf{z}-\mathbf{y})}{\lvert{\mathbf{x}-\mathbf{y}}\rvert\lvert{\mathbf{z}-\mathbf{y}}\rvert}$

The angle is independent of this choice, because the inner product is determined by the quadratic form.

The above is what is written in the book.

I can't understand why the angle is independent of the choice of coordinates(bijection) and the meaning of 'inner product is determined by the quadratic form.'.

Could you explain them?

Answer 1

Suppose you change the coordinate system, then

$ x = O' + R x' $

$ y = O'+ R y' $

$ z = O' + R z' $

where $O'$ is the position of the origin of the $O'x'y'z'$ reference frame when expressed in the $Oxyz$ system, and $R$ is a rotation matrix. it follows that

$ x- y = R (x' - y') $

$ z - y = R (z' - y') $

Then

$ (x - y) \cdot (z - y) = (x - y)^T (z - y) \\= (x' - y')^T R^T R (z' - y') = (x' - y')^T (z' - y') = (x' - y') \cdot (z' - y') $

And for the same reason,

$ \| x - y \| = \sqrt{ (x - y) \cdot (x - y) } = \sqrt{ (x' - y') \cdot (x' - y') } $

and

$ \| z - y \| = \sqrt{ (z - y) \cdot (z - y) } = \sqrt{ (z' - y') \cdot (z' - y') } $

Therefore,

$ \dfrac{ (x - y) \cdot (z - y) }{\| x - y \| \| z- y\|} = \dfrac{ (x' - y') \cdot (z' - y') }{ \| x' - y' \| \| z' - y' \| } $

Hence, the cosine of the angle $\angle xyz $ will be the same when computed using any coordinate system.

Answer 2

The (bilinear) form defining the Euclidean structure is a function from vectors to real numbers. No matter how you express the vectors, their inner product is uniquely defined. If you change coordinates, the expression (matrix) of the form changes, but not the output.

The norm is recovered uniquely from the inner product. It is just a particular case: $\|v\|^2=v\cdot v$.

Therefore, neither the numerator or the denominator in the formula for the cosine change when you change coordinates. That must be the case if the angle is going to have a geometric meaning. Geometric objects should be invariant by changes of coordinates.