Question

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

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

Answer 1

$1)$ Using section formula
Given points are $A(-2,3,5)\&B(1,-4,6)$ Consider the points $P(x,y,z,)$ that divides line in ratio $2:3$ internally.

We know that, coordinates of point that divides line segment joining $(x_1,y_1,z_1)\&(x_2,y_2,z_2)$ in the ratio $m:n$ internally is,

$(x,y,z)$

$=(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n},\frac{m{z}_2+n{z}_1}{m+n})$ $...\left(i\right)$


Here,

$x_1=-2,y_1=3,z_1=5$

$x_2=1,y_2=-4,z_2=6$

$m=2,n=3$

Putting these values in equation $\left(i\right),$

Co-ordinate of point $P$ is
$P(x,y,z)=(\frac{2\left(1\right)+3(-2)}{2+3},\frac{2(-4)+3\left(3\right)}{2+3},\frac{2\left(6\right)+3\left(5\right)}{2+3})$

$=(\frac{2-6}{5},\frac{-8+9}{5},\frac{12+15}{5})$

$=(\frac{-4}{5},\frac{1}{5},\frac{27}{5})$

$2)$ Using section formula
Given points are $A(-2,3,5)\&B(1,-4,6)$ Consider the points $P(x,y,z,)$ that divides line in ratio $2:3$ extermally.

We know that, coordinates of point that divides line segment joining $(x_1,y_1,z_1)\&(x_2,y_2,z_2)$ in the ratio $m:n$ extermally is,

$(x,y,z)$

$=(\frac{m{x}_2-n{x}_1}{m-n},\frac{m{y}_2-n{y}_1}{m-n},\frac{m{z}_2-n{z}_1}{m-n})$ $...\left(i\right)$



Here,

$x_1=-2,y_1=3,z_1=5$

$x_2=1,y_2=-4,z_2=6$

$m=2,n=3$

Putting these values in equation $\left(i\right),$ then we get

$(x,y,z)=(\frac{2\left(1\right)-3(-2)}{2-3},\frac{2(-4)-3\left(3\right)}{2-3},\frac{2\left(6\right)-3\left(5\right)}{2-3})$

$=(\frac{2+6}{-1},\frac{-8-9}{-1},\frac{12-15}{-1})$

$=(\frac{8}{-1},\frac{-17}{-1},\frac{-3}{-1})$

$=(-8,17,3)$