INTRODUCTION TO THREE DIMENSIONAL GEOMETRY — MATHS STD 11 Science — Question

Let R and S be two points which trisect the joining of P and Q.
PR = RS = SQ

Point R divide the join of PQ in the ratio 1:2.
$\therefore$ Coordinates of R are
$\left[ \frac { 1 ( 10 ) + 2 ( 4 ) } { 1 + 2 } , \frac { 1 ( - 16 ) + 2 ( 2 ) } { 1 + 2 } , \frac { 1 ( 6 ) + 2 ( - 6 ) } { 1 + 2 } \right]$
$= \left( \frac { 10 + 8 } { 3 } , \frac { - 16 + 4 } { 3 } , \frac { 6 - 12 } { 3 } \right) = \left( \frac { 18 } { 3 } , \frac { - 12 } { 3 } , \frac { - 6 } { 3 } \right)$
= (6, -4, -2)
Also, point S divides the join of PQ in the ratio 2:1.
$\therefore$ Coordinates of S are
$\left[ \frac { 2 ( 10 ) + 1 ( 4 ) } { 1 + 2 } , \frac { 2 ( - 16 ) + 1 ( 2 ) } { 1 + 2 } , \frac { 2 ( 6 ) + 1 ( - 6 ) } { 1 + 2 } \right]$
$= \left( \frac { 20 + 4 } { 3 } , \frac { - 32 + 2 } { 3 } , \frac { 12 - 6 } { 3 } \right)$
$= \left( \frac { 24 } { 3 } , \frac { - 30 } { 3 } , \frac { 6 } { 3 } \right) $= ( 8 , - 10,2 )