Find the coordinates of the point which divides the line segment joining the points (–2, 3, 5) and (1, –4, 6)
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.

Solution:

(i) The coordinates of point $R$ that divides the line segment joining points $P\left(x_{1}, y_{1}, z_{1}\right)$ and $Q\left(x_{2}, y_{2}, z_{2}\right)$ internally in the ratio $m: n$ are

$\left(\frac{m x_{2}+n x_{1}}{m+n}, \frac{m y_{2}+m y_{1}}{m+n}, \frac{m z_{2}+n z_{1}}{m+n}\right)$

Let R (x, yz) be the point that divides the line segment joining points(–2, 3, 5) and (1, –4, 6) internally in the ratio 2:3

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

i.e., $x=\frac{-4}{5}, y=\frac{1}{5}$, and $z=\frac{27}{5}$

Thus, the coordinates of the required point are $\left(-\frac{4}{5}, \frac{1}{5}, \frac{27}{5}\right)$.

(ii) The coordinates of point $R$ that divides the line segment joining points $P\left(x_{1}, y_{1}, z_{1}\right)$ and $Q\left(x_{2}, y_{2}, z_{2}\right)$ externally in the ratio $m$ : $n$ are

$\left(\frac{m x_{2}-n x_{1}}{m-n}, \frac{m y_{2}-n y_{1}}{m-n}, \frac{m z_{2}-n z_{1}}{m-n}\right)$

Let R (x, yz) be the point that divides the line segment joining points(–2, 3, 5) and (1, –4, 6) externally in the ratio 2:3

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

i.e., $x=-8, y=17$, and $z=3$

Thus, the coordinates of the required point are $(-8,17,3)$.