Question

Find the coordinates of the point which divides the line segment joining the points $(-2,3,5)$ and $(1,-4,6)$ in the ratio. (i) 2 : 3 internally (ii) 2 : 3 externally.

Answer 1

(i) Let $P(x,y,z)$ be any point which divides the line segment joining points $A(-2,3,5)$ and $B(1,-4,6)$ in the ratio 2 : 3 internally.

Then

$x=\frac{2\times 1+3\times -2}{2+3}=\frac{2-6}{5}=\frac{-4}{5}$

$y=\frac{2\times -4+3\times 3}{2+3}=\frac{-8+9}{5}=\frac{1}{5}$

$z=\frac{2\times 6\times 3\times 5}{2+3}=\frac{12+15}{5}=\frac{27}{5}$

$\therefore$ Coordinates of P are $(\frac{-4}{5},\frac{1}{5},\frac{27}{5})$

(ii) Let $P(x,y,z)$ be any point which divides the line segment joining points $A(-2,3,5)$ and $B(1,-4,6)$ in the ratio 2 : 3 externally.

Then $x=\frac{2\times 1+(-3)\times x-2}{2+(-3)}=\frac{2+6}{-1}=-8$

$y=\frac{2\times -4+(-3)\times 3}{2+(-3)}=\frac{-8-9}{-1}=17$

$z=\frac{2\times 6_{(}-3)\times 5}{2+(-3)}=\frac{12-15}{-1}=3$

$\therefore$ Coordinates of P are $(-8,17,3).$