Question

Find the coordinates of the points which trisect the line segment $PQ$ formed by joining the points $P(1,1,-1)$ and $Q(5,-11,0)$

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.

$\therefore A=(\frac{1\times 5+2\times 1}{1+2},\frac{1\times -11+2\times 1}{1+2},\frac{1\times 0+2\times -1}{1+2})$

$\therefore A=(\frac{5+2}{3},\frac{-11+2}{3},\frac{0-2}{3})$

$\therefore A=(\frac{7}{3},\frac{-9}{3},\frac{-2}{3})$

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

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.

$\therefore B=(\frac{2\times 5+1\times 1}{2+1},\frac{2\times -11+1\times 1}{2+1},\frac{2\times 0+1\times -1}{2+1})$

$=(\frac{10+1}{3},\frac{-22+1}{3},\frac{0-1}{3})$

$=(\frac{11}{3},\frac{-21}{3},\frac{-1}{3})$

$\therefore B=(\frac{11}{3},-7,\frac{-1}{3})$