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

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.

It is given that, C divides the line segment joining A and B in the ratio 3 : 1.

$$\begin{gathered} 11\hat{i}-3\hat{j}=\frac{3\times \left(12\hat{i}+\mu \hat{j}\right)+1\times \left(\lambda \hat{i}+3\hat{j}\right)}{3+1} \\ \Rightarrow 11\hat{i}-3\hat{j}=\frac{\left(36+\lambda \right)\hat{i}+\left(3\mu +3\right)\hat{j}}{4} \\ \Rightarrow 44\hat{i}-12\hat{j}=\left(36+\lambda \right)\hat{i}+\left(3\mu +3\right)\hat{j} \end{gathered}$$
Equating the corresponding components, we get

$36+\lambda =44$

$\Rightarrow \lambda =44-36=8$

and

$3\mu +3=-12$

$\Rightarrow 3\mu =-12-3=-15$

$\Rightarrow \mu =-5$

Thus, the values of $\lambda$ and $\mu$ are 8 and −5, respectively.