Question

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 \[\overrightarrow{a} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k} \text{ and }\overrightarrow{b} = - \hat{i} + \hat{j} + \hat{k} .\]

Answer 1

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