Question

The position vectors of points A, B and C are $\lambda \hat{i}+3\hat{j},12\hat{i}+\mu \hat{j}$ and $11\hat{i}-3\hat{j}$ respectively. If C divides the line segment joining A and B in the ratio $3:1$, find the values of $\lambda$ and $\mu$.

Answer 1

$\begin{array}{c}\overrightarrow{OA}=\lambda \hat{i}+3\hat{j}\\ \\ \overrightarrow{OB}=12\hat{i}+\mu \hat{j}\\ \\ \overrightarrow{OC}=11\hat{i}-3\hat{j}\end{array}$

Since, $C$ divides $AB$ in the ratio $3:1$

Therefore,

$\overrightarrow{OC}=\frac{3\overrightarrow{OB}+\overrightarrow{OA}}{3+1}$

$4\overrightarrow{OC}=3\overrightarrow{OB}+\overrightarrow{OA}$

$4(11\hat{i}-3\hat{j})=3(12\hat{i}-\mu \hat{j})+(\lambda \hat{i}+3\hat{j})$

$44\hat{i}-12\hat{j}=(36+\lambda )\hat{i}+(3\mu +3)\hat{j}$

$36+\lambda 44$

$\lambda =8$

And

$3\mu +3=-12$

$\mu =-5$

Therefore,

$\lambda =8$ and $\mu =-5$