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Q.
Find the coordinates of the points which trisect the line segment joining the points $P(4, 2, -6)$ and $Q(10, -16,6)$.
Introduction to Three Dimensional Geometry
Solution:
Let $B$ and $S$ be two points which trisect the join of $PQ$.
$\because$ Point $R$ divides the join of $PQ$ in the ratio $1 :2$
$\therefore $ Coordinates of $B$ is
$\left(\frac{1\times10+2\times4}{1+2}, \frac{1\times \left(-16\right)+2\times2}{1+2}, \frac{1\times 6+2\times \left(-6\right)}{1+2}\right)$
$= \left(6, -4,-2\right)$
Also point $S$ divides the join of $PQ$ in the ratio $2: 1$
$\therefore $ Coordinates of $R$ is
$\left(\frac{2\times 10+1\times 4}{1+2}, \frac{2\times \left(-16\right)+1\times 2}{1+2}, \frac{2\times 6+1\times \left(-6\right)}{1+2}\right)$
$=\left(8, -10, 2\right)$.
