The three point $A (0, 0, 0), B (2, - 3, 3), C (- 2, 3, - 3)$ are collinear. Find in what ratio each point divides the segment joining other two.

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Solution

Let us assume that point B divides the line AC in the ratio k:1
$L e t (x_{1}, y_{1}, z_{1}) = (0, 0, 0) (x, y, z) = (2, - 3, 3) (x_{2}, y_{2}, z_{2}) = (- 2, 3, - 3) B y s e c t i o n f o r m u l a : x = \frac{m_{1} x_{2} + m_{2} x_{1}}{m_{1} + m_{2}} \Rightarrow 2 = \frac{k . (- 2) + 1. (0)}{k + 1} \Rightarrow k = \frac{- 1}{4}$
on neglecting the negative sign we get,
therefore ratio is 1:4

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