Find the coordinates of the point which divides the line segment joining the points (–2, 3, 5) and (1, –4, 6)
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.
(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, 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
$x=\frac{2(1)+3(-2)}{2+3}, y=\frac{2(-4)+3(3)}{2+3}$, 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, 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
$x=\frac{2(1)-3(-2)}{2-3}, y=\frac{2(-4)-3(3)}{2-3}$, 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 $(-8,17,3)$.
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