Find the coordinate of the points which trisect

Given the line segment joining the points are $A(2,1,-3)$ and $B(5,-8,3)$. Now let $\mathrm{P}\left(\mathrm{x}_{1}, \mathrm{y}_{1}, \mathrm{z}_{1}\right)$ and $\mathrm{Q}\left(\mathrm{x}_{2}, \mathrm{y}_{2}, \mathrm{z}_{2}\right)$ be the points which trisects the line segment.

$\Rightarrow \mathrm{P}$ divides $\mathrm{AB}$ in the ratio $2: 1$

$\Rightarrow x_{1}=\frac{2+2 \times 5}{1+2}=4$

$\Rightarrow y_{1}=\frac{1+2 \times(-8)}{1+2}=-5$

$\Rightarrow \mathrm{z}_{1}=\frac{-3+2 \times 3}{1+2}=1$

$\Rightarrow \mathrm{Q}$ divides $\mathrm{AP}$ in the ratio $1: 1$

$\Rightarrow \mathrm{x}_{2}=\frac{2+4}{2}=3$

$\Rightarrow y_{2}=\frac{1+(-5)}{2}=-2$

$\Rightarrow \mathrm{z}_{2}=\frac{-3+1}{2}=-1$

$\therefore(4,-5,1)$ and $(3,-2,-1)$ are the coordinate of the points which trisect the line segment joining the points $A(2,1,-3)$ and $B(5,-8,3)$.