\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{calc}
\begin{document}
\begin{tikzpicture}
%zuerst ein paar styles definieren
[
    help/.style={->,color=gray,very thin},
    nodehelp/.style={fill=white,above=1pt,midway},
    imp/.style={->,very thick},
    pro/.style={color=green!50!blue},
    pronode/.style={anchor=south east, fill=white, midway}]
%Basiskoordinaten
\coordinate (null) at (0cm,0cm);
\coordinate (vec) at (-3cm,4cm);
%path für let operator
\path
 let
    \p1 = (vec),
    \n1 = {veclen(\x1,\y1)}
 in
   coordinate (shift) at (\n1,0cm);
%abgeleitete koordinaten
\coordinate (mirror) at ($(vec)-(shift)$);
\coordinate (omirror) at ($(vec)+(shift)$);
\coordinate (mirpro) at ($(null)!(vec)!(mirror)$);
\coordinate (omirpro) at ($(null)!(vec)!(omirror)$);
\coordinate (arc) at ($ 0.1*(vec)-0.1*(omirpro)$);
%Koordinatensystem zeichnen
\draw[->,shorten <=1cm,shorten >=1cm] (-10cm,0cm)--(10cm,0cm);
\draw[->] (0cm,-1.5cm) -- (0cm,6cm)
    node[anchor=south,at end] {\huge{Householder Spiegelung}};
%hilfslinien unauffällig machen
\draw[color=gray, very thin] (shift)
 let
    \p1 = (shift),
    \n1 = {veclen(\x1,\y1)}
 in
     arc(0:180:\n1);
\draw[help] (shift) -- (omirror);
\draw[help] (vec) -- (mirror)
    node[nodehelp] {${}-\|v\|\cdot  e_1$};
\draw[help] (vec) -- (omirror)
    node[nodehelp] {$\|v\|\cdot e_1$};
\draw[help] (null) -- ($ 1.1*(mirror)$)
    node[anchor=south] {\large{Spiegelebene}};
\draw (vec) -- (omirpro);
\draw ($(omirpro) + (arc)$)
 let
    \p1 = (arc),
    \n1 = {veclen(\x1,\y1)},
    \n2 = {atan2(\x1,\y1)}
 in
    arc(\n2:\n2+90:\n1);
\fill[color=black] (omirpro)
 let
    \p1 = ($0.5*(arc)$),
    \n1 = {veclen(\x1,\y1)},
    \n2 = {0.125*veclen(\x1,\y1)},
    \n3 = {atan2(\x1,\y1)}
 in
  ++(\n3+45:\n1)
    circle (\n2);
\draw ($(mirpro) + (arc) $)
 let
    \p1 = (arc),
    \n1 = {veclen(\x1,\y1)},
    \n2 = {atan2(\x1,\y1)}
 in
    arc(\n2:\n2+90:\n1);
\fill[color=black] (mirpro)
 let
    \p1 = ($0.5*(arc)$),
    \n1 = {veclen(\x1,\y1)},
    \n2 = {0.125*veclen(\x1,\y1)},
    \n3 = {atan2(\x1,\y1)}
 in
  ++ (\n3+45:\n1)
    circle (\n2);
\draw[color=blue!50!green, |<->|] ($(null)-0.01*(mirror)$) -- ($(omirpro)- 0.01*(mirror)$)
    node[anchor=north west, fill=white, midway] {$\alpha:=\langle \tilde{\omega}, v \rangle$};
%wichtige linien, zum schluss zeichnen
\draw[imp,color=blue] (null) -- (vec)
    node[anchor=south west,midway] {$v$};
\draw[imp,color=green] (null) --(mirror)
    node[anchor=north east, near end] {$v-\|v\| \cdot e_1$};
\draw[imp,color=green] (null) -- (omirror)
    node[anchor=north west,fill=white, near end,align=center] {$\omega:=v+\|v\| \cdot e_1$\\
                                                    $\tilde{\omega}=\frac{\omega}{\|\omega\|}$};
\draw[imp,pro] (vec) -- (mirpro)
    node[pronode] {$-\alpha \tilde{\omega}$};
\draw[imp,pro, ] (mirpro) -- ($-1*(shift)$)
    node[pronode] {$-\alpha \tilde{\omega}$};%there is a small place between the lines we draw it
\draw[pro, very thick,shorten >= 4mm] (vec) -- ($-1*(shift)$); %seperate to have a consistent node placing
\draw[imp,color=blue!40!green] (null)-- ($ -1*(shift)$)
    node[fill=white, below=3pt,at end] {$Q_\omega \cdot v=-\|v\|\cdot e_1$};
\end{tikzpicture}
\end{document}