Find the coordinates of the points which divides the line joining the points $(2, - 4, 3), (- 4, 5, - 6)$ in the ratio
$(i) 1 : - 4$
$(i i) 2 : 1$

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Solution

(i) The coordinates of the point which divides the line joining the point (2,-4,3) and (-4,5,-6) in the ratio 1:-4 is given by
$x = \frac{m_{1} x_{2} + m_{2} x_{1}}{m_{1} + m_{2}} = \frac{1. (- 4) + (- 4) .2}{1 + (- 4)} = 4 y = \frac{m_{1} y_{2} + m_{2} y_{1}}{m_{1} + m_{2}} = \frac{1.5 + (- 4) . (- 4)}{1 + (- 4)} = - 7 z = \frac{m_{1} z_{2} + m_{2} z_{1}}{m_{1} + m_{2}} = \frac{1. (- 6) + (- 4) .3}{1 + (- 4)} = 6 ∴ T h e c o o r d i n a t e i s (4, - 7, 6)$
(ii) The coordinates of the point which divides the line joining the point (2,-4,3) and (-4,5,-6) in the ratio 2:1 is given by
$x = \frac{m_{1} x_{2} + m_{2} x_{1}}{m_{1} + m_{2}} = \frac{2. (- 4) + 1.2}{2 + 1} = - 2 y = \frac{m_{1} y_{2} + m_{2} y_{1}}{m_{1} + m_{2}} = \frac{2.5 + 1. (- 4)}{2 + 1} = 2 z = \frac{m_{1} z_{2} + m_{2} z_{1}}{m_{1} + m_{2}} = \frac{2. (- 6) + 1.3}{2 + 1} = - 3 ∴ T h e c o o r d i n a t e i s (- 2, 2, - 3)$

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Find the coordinates of the points which divides the line joining the points (2,−4,3),(−4,5,−6) in the ratio (i)1:−4(ii)2:1

Find the coordinates of the points which divides the line joining the points $(2, - 4, 3), (- 4, 5, - 6)$ in the ratio
$(i) 1 : - 4$
$(i i) 2 : 1$