Question

Find the coordinates of the points which trisect the line segment $PQ$ formed by joining the points $P(2,2,3)$ and $Q(3,9,0)$

Answer 1

Let $A$ and $B$ be the points of trisection of the segment $PQ$, then

$PA=AB=BQ\Rightarrow 2PA=AQ$

$\Rightarrow \frac{PA}{AQ}=\frac{1}{2}$

$\Rightarrow A$ divides the line segment $PQ$ in the ratio $1:2$ internally.

$\therefore A=(\frac{1\times 3+2\times 2}{1+2},\frac{1\times 9+2\times 2}{1+2},\frac{1\times 0+2\times 3}{1+2})$

$\therefore A=(\frac{3+4}{3},\frac{9+4}{3},\frac{0+6}{3})$

$\therefore A=(\frac{7}{3},\frac{13}{3},\frac{6}{3})$

$\therefore A=(\frac{7}{3},\frac{13}{3},2)$

Also,$PA=AB=BQ\Rightarrow PB=2BQ$

$\Rightarrow \frac{PB}{BQ}=\frac{2}{1}$

$\Rightarrow B$ divides the line segment $PQ$ in the ratio $2:1$ internally.

$\therefore B=(\frac{2\times 3+1\times 2}{2+1},\frac{2\times 9+1\times 2}{2+1},\frac{2\times 0+1\times 3}{2+1})$

$=(\frac{6+2}{3},\frac{18+2}{3},\frac{0+3}{3})$

$=(\frac{8}{3},\frac{20}{3},\frac{3}{3})$

$\therefore B=(\frac{8}{3},\frac{20}{3},1)$