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 $\boxed{\mathbf{v}}$