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  4. If the points A ≡(1,2,3), B ≡(3,4,7) and C ≡(-3,-2,-5), then the ratio in which point C divides AB, is

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Q. If the points $A \equiv(1,2,3), B \equiv(3,4,7)$ and $C \equiv(-3,-2,-5)$, then the ratio in which point $C$ divides $AB$, is

Vector Algebra

Solution:

Let $C$ divide $A B$ in the ratio $k$ : 1 then $C(-3,-2,-5) \equiv$
$\left(\frac{3 k +1}{ k +1}, \frac{4 k +2}{ k +1}, \frac{7 k +3}{ k +1}\right)$
$\Rightarrow \frac{3 k +1}{ k +1}=-3, \frac{4 k +2}{ k +1}=-2$ and $\frac{7 k +3}{ k +1}=-5$
$\Rightarrow k =-\frac{2}{3}$ from all relations
Hence, $C$ divides $AB$ extenally in the ratio $2: 3$.