Question

Let point $\overrightarrow{OC}=x\hat{i}+y\hat{j}+z\hat{k}$ lies on line joining points $\overrightarrow{OA}=3\hat{i}-2\hat{j}+\hat{k}$ and $\overrightarrow{OB}=-2\hat{i}+\hat{j}-3\hat{k},$ then which of the following statement(s) is/are correct ?

• A. If $x=y;\overrightarrow{OC}$ divides $\overrightarrow{AB}$ internally in ratio $5:3$
• B. If $y=z;\overrightarrow{OC}$ divides $\overrightarrow{AB}$ internally in ratio $3:4$
• C. If $z=x;\overrightarrow{OC}$ divides $\overrightarrow{AB}$ externally in ratio $2:1$
• D. If $z=x;\overrightarrow{OC}$ divides $\overrightarrow{AB}$ internally in ratio $2:1$

Answer 1

Answer: (A), (B), (C)

If $z=x;\overrightarrow{OC}$ divides $\overrightarrow{AB}$ externally in ratio $2:1$

Given points points $\overrightarrow{OA}=3\hat{i}-2\hat{j}+\hat{k}$ and $\overrightarrow{OB}=-2\hat{i}+\hat{j}-3\hat{k}$
Let point $C$ divides the $\overrightarrow{AB}$ in $\lambda :1$
$\overrightarrow{OC}=\frac{\overrightarrow{OA}+\lambda \overrightarrow{OB}}{\lambda +1}\Rightarrow x\hat{i}+y\hat{j}+z\hat{k}=\frac{(-2\lambda +3)\hat{i}+(\lambda -2)\hat{j}+(-3\lambda +1)}{\lambda +1}$
equating the cofficients of like vectors :
$\Rightarrow x=\frac{-2\lambda +3}{\lambda +1},y=\frac{\lambda -2}{\lambda +1},z=\frac{-3\lambda +1}{\lambda +1}$
For $x=y:-2\lambda +3=\lambda -2$
$\lambda =\frac{5}{3}$
So, division is internally in $5:3$
For $y=z:\lambda -2=-3\lambda +1$
$\Rightarrow \lambda =\frac{3}{4}$
So, division is internally in ratio $3:4$
For $x=z:-3\lambda +1=-2\lambda +3$
$\Rightarrow \lambda =-2$
(negative ratio indicates division is externally)
So, division is externally in ratio $2:1$