Write the position vector of a point dividing the line segment joining points A and B with position vectors $\vec{a}$ and $\vec{b}$ externally in the ratio 1 : 4, where $\vec{a} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k} and \vec{b} = - \hat{i} + \hat{j} + \hat{k} .$

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Solution

The position vectors of $A$ and $B$ are
$\vec{a} = 2 \hat{i} + 3 j + 4 \hat{k} \vec{b} = - \hat{i} + \hat{j} + \hat{k}$
Let $C$ divides $A B$ in the ratio such that AB : CB = 1 : 4
Position vector of $C$ = $\frac{1 (- \hat{i} + \hat{j} + \hat{k}) - 4 (2 \hat{i} + 3 \hat{j} + 4 \hat{k})}{1 - 4}$
= $\frac{- \hat{i} + \hat{j} + \hat{k} - 8 \hat{i} - 12 \hat{j} - 16 \hat{k}}{- 3}$
= $\frac{- 9 \hat{i} - 11 \hat{j} - 15 \hat{k}}{- 3}$
= $3 \hat{i} + \frac{11 \hat{j}}{3} + 5 \hat{k}$

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