Question

Write the position vector of a point dividing the line segment joining points having position vectors \[\hat{i} + \hat{j} - 2 \hat{k} \text{ and }2 \hat{i} - \hat{j} + 3 \hat{k}\] externally in the ratio 2:3.

Answer 1

Let A and B be the points with position vectors \[\vec{a} = \hat{i} + \hat{j} - 2 \hat{k} , \vec{b} = 2 \hat{i} - \hat{j} + 3 \hat{k}\] respectively.
Let C divide AB  externally in the ratio 2 : 3 such that AC : CB = 2 : 3
∴ Position vector of C =\[\frac{2\left( 2 \hat{i} - \hat{j} + 3 \hat{k} \right) - 3\left( \hat{i} + \hat{j} - 2 \hat{k} \right)}{2 - 3}\]
= \[\frac{4 \hat{i} - 2 \hat{j} + 6 \hat{k} - 3 \hat{i} - 3 \hat{j} + 6 \hat{k}}{- 1}\]
= \[\frac{\hat{i} - 5 \hat{j} + 12 \hat{k}}{- 1}\]
=  \[- \hat{i} + 5 \hat{j} - 12 \hat{k}\]