Question

Find the coordinates of the points which trisect the line segment joining the points $A(2,1,–3)$ and $B(5,–8,3).$

Answer 1

Identifying the ratio in which a point divides the line joining two points.

Let $P$ and $Q$ be points of trisection Therefore,$AP=PQ=QB$


$\therefore$ P divides AB in 1 : 2 and Q divides AB in the ratio 2 : 1.

Applying the section formula

The coordinate of the point R which divides the line segment joining two points $P(x_1,y_2,z_3)$ and $Q(x_2,y_2,z_2)$ internally in the ratio m : n is given by

$R(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n},\frac{m{z}_2+n{z}_1}{m+n})$

$\therefore P=\left({}_{\frac{1\times \left(3\right)+2\times (-3)}{3}}^{\frac{1\times 5+2\times 2}{3},\frac{1\times (-8)+2\left(1\right)}{3},}\right)$

$\Rightarrow P=(3,–2,–1)$

$Q=\left({}_{\frac{2\times \left(3\right)+1\times (-3)}{3}}^{\frac{2\times 5+1\times 2}{3},\frac{2\times (-8)+1\left(1\right)}{3},}\right)$

$\Rightarrow Q=(4,–5,1)$

Therefore, points of trisection are $P(3,–2,–1)$ and $Q(4,–5,1)$