Question

Find the coordinates of the points which trisect the line segment joining the points $P(4,2,-6)$ and $Q(10,-16,6)$

Answer 1

Given coordinates of $P$ as $(4,2,-6)$ and $Q$ as $(10,-16,6)$

Let $R$ and $S$ be the points of trisection of line segment $PQ$.

Then $R$ divides $PQ$ in the ratio $1:2$ and $S$ is the mid-point of $RQ$.

Let the coordinates of $R$ be $(x,y,z)$

By using section formula,

$x=\frac{1\left(10\right)+2\left(4\right)}{1+2},y=\frac{1(-16)+2\left(2\right)}{1+2},z=\frac{1\left(6\right)+2(-6)}{1+2}$

$\Rightarrow x=\frac{18}{3},y=\frac{-12}{3},z=\frac{-6}{3}$

$\Rightarrow x=6,y=-4,z=-2$

So, the coordinates of $R$ are $(6,-4,-2)$.

Now, since $S$ is the mid-point of $RQ$.