Find the coordinates of the point which trisect $A B$ given that $A (2, 1, - 3)$ and $B (5, - 8, 3)$ .

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Solution

line AB is divided into three equal parts i.e, AC,CD,DB
for point C the ratio we take is 1:2
$B y s e c t i o n f o r m u l a : x = \frac{m_{1} x_{2} + m_{2} x_{1}}{m_{1} + m_{2}} = \frac{1.5 + 2.2}{1 + 2} = 3 y = \frac{m_{1} y_{2} + m_{2} y_{1}}{m_{1} + m_{2}} = \frac{1 (- 8) + 2.1}{1 + 2} = - 2 z = \frac{m_{1} z_{2} + m_{2} z_{1}}{m_{1} + m_{2}} = \frac{1.3 + 2 (- 3)}{1 + 2} = - 1$
for point D the ratio we take is 2:1
$B y s e c t i o n f o r m u l a : x = \frac{m_{1} x_{2} + m_{2} x_{1}}{m_{1} + m_{2}} = \frac{2.5 + 1.2}{2 + 1} = 4 y = \frac{m_{1} y_{2} + m_{2} y_{1}}{m_{1} + m_{2}} = \frac{2 (- 8) + 1.1}{2 + 1} = - 5 z = \frac{m_{1} z_{2} + m_{2} z_{1}}{m_{1} + m_{2}} = \frac{2.3 + 1 (- 3)}{2 + 1} = 1$
therefore coordinates are $(3, - 2, - 1)$ and $(4, - 5, 1)$

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Find the coordinates of the points which trisect the line segment AB, if two points are A(2, 1, -3) and B (5, -8, 3).

Let A(2, 1, -3) and B(5, -8, 3) be two given points. Find the coordinates of the points of trisection of the line segment AB.

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