Linear mapping/Angle-preserving/All angles/Remark

For an angle-preserving mapping

${\displaystyle \varphi \colon V\longrightarrow W}$

between Euclidean vector spaces ${\displaystyle {}V}$ and ${\displaystyle {}W}$, not only the angles at the origin, but all angles are preserved. For points ${\displaystyle {}P,Q,R\in V}$, the angle of the triangle at ${\displaystyle {}Q}$ coincides with the angle at ${\displaystyle {}\varphi (Q)}$ of the image triangle ${\displaystyle {}\varphi (P),\varphi (Q),\varphi (R)}$, because of

${\displaystyle {}\angle (P,Q,R)=\angle ({\overrightarrow {QP}},{\overrightarrow {QR}})=\angle (\varphi ({\overrightarrow {QP}}),\varphi ({\overrightarrow {QR}}))=\angle ({\overrightarrow {\varphi (Q)\varphi (P)}},{\overrightarrow {\varphi (Q)\varphi (R)}})=\angle (\varphi (P),\varphi (Q),\varphi (R))\,.}$