Question

Find the coordinates of the point which divides the join of $P(2,-1,4)$ and $Q(4,3,2)$ in the ratio $2:3$ (i) internally (ii) 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 points that divides the line segment joining points $(2,-1,4)$ and $(4,3,2)$ internally in the ratio $2:3$

$x=\frac{2\times 4+3\times 2}{2+3},y=\frac{2\times 3+3\times (-1)}{2+3},z=\frac{2\times 2+3\times 4}{2+3}$

$\Rightarrow x=\frac{8+6}{5},y=\frac{6-3}{5},z=\frac{4+12}{5}$

$\Rightarrow x=\frac{14}{5},y=\frac{3}{5},z=\frac{16}{5}$​

Thus, the coordinates of the required point are $(\frac{14}{5},\frac{3}{5},\frac{16}{5})$

(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,-1,4) and (4,3,2) externally in the ratio $2:3$

$x=\frac{2\times 4-3\times 2}{2-3},y=\frac{2\times 3-3\times (-1)}{2-3},z=\frac{2\times 2-3\times 4}{2-3}$
$\Rightarrow x=\frac{8-6}{-1},y=\frac{6+3}{-1},z=\frac{4-12}{-1}$​

i.e., $x=-2,y=-9,$ and $z=8$

Thus, the coordinates of the required point are $(-2,-9,8).$