Question

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

Answer 1

Consider the problem

Let,

the points of intersection

$P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$

then,

$P$ divides $AB$ in the ratio $1:2$

And

$Q$ divides $AB$ in the ratio $2:1$

therefore,

$(x_1,y_1,z_1)=(\frac{5+4}{3},\frac{-8+2}{3},\frac{3-6}{3})$

$(x_1,y_1,z_1)=(3,-2,-1)$

$(x_2,y_2,z_2)=(\frac{10+2}{3},\frac{-16+1}{3},\frac{6-3}{3})$

$(x_2,y_2,z_2)=(4,-5,1)$

therefore,

Points which trisects the line $AB$ are $(3,-2,-1)$ and $(4,-5,-1)$