Next choose v1 to have all but the first two elements zero.
\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} -\frac{1}{3} \\ \frac{\sqrt{8}}{3} \\ 0 \end{pmatrix}, \begin{pmatrix} -\frac{1}{3} \\ y_2 \\ z_2 \end{pmatrix}, \begin{pmatrix} -\frac{1}{3} \\ y_3 \\ z_3 \end{pmatrix} .