I am really frustrated because i just cant find my error.
Let $\quad v_1 = \begin{pmatrix}1 \\0\\i \end{pmatrix} , v_2 = \begin{pmatrix}0 \\-i\\1 \end{pmatrix} , v_3 = \begin{pmatrix}i \\1\\0 \end{pmatrix}$
First step
$ s_1 = \frac{v_1}{|v_1|} = \frac{1}{\sqrt2}\cdot\begin{pmatrix}1 \\0\\i \end{pmatrix}$
Second step
$w_2 = v_2 -<v_2,s_1>\cdot s_1 = \begin{pmatrix}0 \\-i\\1 \end{pmatrix} - <\begin{pmatrix}0 \\-i\\1 \end{pmatrix},\frac{1}{\sqrt2}\cdot\begin{pmatrix}1 \\0\\i \end{pmatrix}>\frac{1}{\sqrt2} \cdot \begin{pmatrix}1 \\0\\i \end{pmatrix} $
$w_2 = \begin{pmatrix}0 \\-i\\1 \end{pmatrix} - \frac{i}{2}\cdot \begin{pmatrix}1 \\0\\i \end{pmatrix} = \frac{1}{2}\begin{pmatrix}-i \\-2i\\3 \end{pmatrix}$
$s_2 = \frac{w_2}{|w_2|} \quad \text{with} \quad |w_2| = \sqrt{\frac{7}{2}} \Longrightarrow s_2 = \frac{1}{\sqrt{14}} \cdot \begin{pmatrix}-i \\-2i\\3 \end{pmatrix}$
According to Wolfram Alpha this is wrong already and i cant find my mistake. I hope you can help me here.