I had a question about the orbit based Gibbs' Method. In the image below is the Gibbs' methods algorithm. I see that it requires an input of three position vectors and it outputs one velocity vector. At what time does the velocity vector represent? Does the velocity vector correspond with position vector r$1,$ r$2,$ or r$3?$ Can you explain a bit about this method?

$3\mathrm{.}1\mathrm{.}4\mathrm{.}1$ Algorithm for Gibbs' Method

Given: $r_i$ for iin$\{1,2,3\}$

Verify that $(\frac{r_1}{|}|r_1||)*(r_2\frac{r_3}{|}|r_2{r}_3||)=0.$

Calculate $n,d,$ and $s$ via

$n=||r_1||(r_2{r}_3)+||r_2||(r_3{r}_1)+||r_3||(r_1{r}_2)$

$d=(r_1{r}_2)+(r_2{r}_3)+(r_3{r}_1)$

$s=r_1[||r_2||-||r_3||]+r_2[||r_3||-||r_1||]+r_3[||r_1||-||r_2||]$

Output: $v_2,$ which is calculated as

$v_2=\sqrt[\phantom{2}]{\frac{}{||n||*||d||}}[\frac{d{r}_2}{||r_2||}+s]$