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) The coordinates of point R that divides the line segment joining points $P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$ internally in the ratio $m:n$ are
$(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n},\frac{m{z}_2+n{z}_1}{m+n})$
Let $R(x,y,z)$ be the point that divides the line segment joining points $(-2,3,5)$ and $(1,-4,6)$ internally in the ratio $2:3$
$x=\frac{2\left(1\right)+3(-2)}{2+3},y=\frac{2(-4)+3\left(3\right)}{2+3}$ and $z=\frac{2\left(6\right)+3\left(5\right)}{2+3}$
i.e., $x=\frac{-4}{5},y=\frac{1}{5}$and $z=\frac{27}{5}$
Thus the coordinates of the required point are $(-\frac{4}{5},\frac{1}{5},\frac{27}{5})$
(ii) The coordinates of point $R$ that divides the line segment joining points $P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$ externally in the ratio $m:n$ are
$(\frac{m{x}_2-n{x}_1}{m-n},\frac{m{y}_2-n{y}_1}{m-n}\frac{m{z}_2-n{z}_1}{m-n})$
Let $R(x,y,z)$ be the point that divides the line segment joining points $(-2,3,5)$ and $(1,-4,6)$ externally in the ratio $2:3$
$x=\frac{2\left(1\right)-3(-2)}{2-3},y=\frac{2(-4)-3\left(3\right)}{2-3}$ and $z=\frac{2\left(6\right)-3\left(5\right)}{2-3}$
i.e.,$x=-8,y=17$ and $z=3$
Thus the coordinates of the required point are $(-8,17,3)$