Question

$\left(i\right)$ Find the coordinates of the point which divides the line segment joining the points $(1,-2,3)\text{and}(3,4,-5)$ in the ratio $2:3$
Internally

$\left(ii\right)$ Find the coordinates of the point which divides the line segment joining the points $(1,-2,3)\text{and}(3,4,-5)$ in the ratio $2:3$
Externally

Answer 1

$\left(i\right)$ Using section formula
Given points are $(1,-2,3)\text{and}(3,4,-5)$
Consider the points $P(x,y,z)$ that divides line segment 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,
$P(x,y,z)$

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

Here, $x_1=1,y_1=-2,z_1=3$
$x_2=3,y_2=4,z_2=-5$
$m=2,n=3$

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

$=(\frac{2\left(3\right)+3\left(1\right)}{2+3},\frac{2\left(4\right)+3(-2)}{2+3},\frac{2(-5)+3\left(3\right)}{2+3})$

$=(\frac{6+3}{5},\frac{8-6}{5},\frac{-10+9}{5})$

$=(\frac{9}{5},\frac{2}{5},\frac{-1}{5})$

$\left(ii\right)$ Using section formula
Given points are $(1,-2,3)\text{and}(3,4,-5)$
Consider the points $P(x,y,z)$ that divides line segment in ratio $2:3$ externally
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 $m:n$
externally is,
$P(x,y,z)$

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

Here, $x_1=1,y_1=-2,z_1=3$
$x_2=3,y_2=4,z_2=-5$
$m=2,n=3$

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

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

$=(\frac{6-3}{(-1)},\frac{8+6}{(-1)},\frac{-19}{(-1)})$

$=(\frac{3}{(-1)},\frac{14}{(-1)},\frac{-19}{(-1)})$

$=(-3,-14,19)$
​​​​​​