Let the given vector be denoted by \(\mathbf{v}\): \[\mathbf{v} = \begin{pmatrix} v_1 \ v_2 \ v_3 \ v_4 \ v_5 \ v_6 \end{pmatrix}\] where \(v_1 = \frac{v_{\alpha x} (y_{\alpha} - y_{\beta})^2 + v_{\alpha x} (z_{\alpha} - z_{\beta})^2 - v_{\alpha y} (x_{\alpha} - x_{\beta})(y_{\alpha} - y_{\beta}) - v_{\alpha z} (x_{\alpha} - x_{\beta})(z_{\alpha} - z_{\beta}) - x_{\alpha} \sqrt{\Delta} + x_{\beta} \sqrt{\Delta}}{D}\)

\(v_2 = \frac{-v_{\alpha x} (x_{\alpha} - x_{\beta})(y_{\alpha} - y_{\beta}) + v_{\alpha y} (x_{\alpha} - x_{\beta})^2 + v_{\alpha y} (z_{\alpha} - z_{\beta})^2 - v_{\alpha z} (y_{\alpha} - y_{\beta})(z_{\alpha} - z_{\beta}) - y_{\alpha} \sqrt{\Delta} + y_{\beta} \sqrt{\Delta}}{D}\)

\(v_3 = \frac{- v_{\alpha x} (x_{\alpha} - x_{\beta})(z_{\alpha} - z_{\beta}) - v_{\alpha y} (y_{\alpha} - y_{\beta})(z_{\alpha} - z_{\beta}) + v_{\alpha z} (x_{\alpha} - x_{\beta})^2 + v_{\alpha z} (y_{\alpha} - y_{\beta})^2 - (z_{\alpha} - z_{\beta})\sqrt{\Delta}}{D}\)

\(v_4 = \frac{v_{\alpha x} (y_{\alpha} - y_{\beta})^2 + v_{\alpha x} (z_{\alpha} - z_{\beta})^2 - v_{\alpha y} (x_{\alpha} - x_{\beta})(y_{\alpha} - y_{\beta}) - v_{\alpha z} (x_{\alpha} - x_{\beta})(z_{\alpha} - z_{\beta}) + x_{\alpha} \sqrt{\Delta} - x_{\beta} \sqrt{\Delta}}{D}\)

\(v_5 = \frac{-v_{\alpha x} (x_{\alpha} - x_{\beta})(y_{\alpha} - y_{\beta}) + v_{\alpha y} (x_{\alpha} - x_{\beta})^2 + v_{\alpha y} (z_{\alpha} - z_{\beta})^2 - v_{\alpha z} (y_{\alpha} - y_{\beta})(z_{\alpha} - z_{\beta}) + y_{\alpha} \sqrt{\Delta} - y_{\beta} \sqrt{\Delta}}{D}\)

\(v_6 = \frac{- v_{\alpha x} (x_{\alpha} - x_{\beta})(z_{\alpha} - z_{\beta}) - v_{\alpha y} (y_{\alpha} - y_{\beta})(z_{\alpha} - z_{\beta}) + v_{\alpha z} (x_{\alpha} - x_{\beta})^2 + v_{\alpha z} (y_{\alpha} - y_{\beta})^2 + (z_{\alpha} - z_{\beta}) \sqrt{\Delta}}{D}\)

where \[\Delta = s_{\beta}^{2} ((x_{\alpha} - x_{\beta})^2 + (y_{\alpha} - y_{\beta})^2 + (z_{\alpha} - z_{\beta})^2) - (v_{\alpha x} (y_{\alpha} - y_{\beta}) + v_{\alpha y} (x_{\alpha} - x_{\beta}) + v_{\alpha z} (z_{\alpha} - z_{\beta}))^2\] and \[D = (x_{\alpha} - x_{\beta})^2 + (y_{\alpha} - y_{\beta})^2 + (z_{\alpha} - z_{\beta})^2\]

We can simplify this further. Let \(\mathbf{r} = (x_{\alpha} - x_{\beta}, y_{\alpha} - y_{\beta}, z_{\alpha} - z_{\beta})\) and \(\mathbf{v}_{\alpha} = (v_{\alpha x}, v_{\alpha y}, v_{\alpha z})\). Then, \[D = |\mathbf{r}|^2 = r_x^2 + r_y^2 + r_z^2\] \[\Delta = s_{\beta}^2 |\mathbf{r}|^2 - (\mathbf{v}_{\alpha} \cdot (\mathbf{r} \times \mathbf{e}))^2\] where \(\mathbf{e} = (1,1,1)\).

Let \(\mathbf{R}_{\alpha} = (x_{\alpha}, y_{\alpha}, z_{\alpha})\) and \(\mathbf{R}_{\beta} = (x_{\beta}, y_{\beta}, z_{\beta})\).

Then, we can write: \[\mathbf{v} = \begin{pmatrix} \frac{\mathbf{v}_{\alpha} \times (\mathbf{r} \times \mathbf{r}) + (\mathbf{R}_{\beta} - \mathbf{R}_{\alpha}) \sqrt{\Delta}}{D} \ \frac{\mathbf{v}_{\alpha} \times (\mathbf{r} \times \mathbf{r}) - (\mathbf{R}_{\beta} - \mathbf{R}_{\alpha}) \sqrt{\Delta}}{D} \end{pmatrix}\] Since \(\mathbf{r} \times \mathbf{r} = \mathbf{0}\), \[\mathbf{v} = \begin{pmatrix} (\mathbf{R}_{\beta} - \mathbf{R}_{\alpha}) \frac{\sqrt{\Delta}}{D} \ - (\mathbf{R}_{\beta} - \mathbf{R}_{\alpha}) \frac{\sqrt{\Delta}}{D} \end{pmatrix}\] Thus, we have \[\mathbf{v} = \begin{pmatrix} -\mathbf{r} \frac{\sqrt{\Delta}}{D} \ \mathbf{r} \frac{\sqrt{\Delta}}{D} \end{pmatrix} = \begin{pmatrix} -\mathbf{r} \frac{\sqrt{\Delta}}{|\mathbf{r}|^2} \ \mathbf{r} \frac{\sqrt{\Delta}}{|\mathbf{r}|^2} \end{pmatrix}\]

Final Answer: The final answer is \(\boxed{\mathbf{v}}\)