Question

The ratio in which $\hat{i}+2\hat{j}+3\hat{k}$ divides the join of $-2\hat{i}+3\hat{j}+5\hat{k}$ and $7\hat{i}-\hat{k}$ is

• A. $1:2$
• B. $1:4$
• C. $2:3$
• D. $3:4$

Answer 1

Answer: (A)

Let the line joining the points with position vectors $-2\hat{i}+3\hat{j}+5\hat{k}$ and $7\hat{i}-\hat{k}$ be divided in the ratio $\lambda :1$ by $\hat{i}+2\hat{j}+3\hat{k}$
$\therefore \frac{\lambda (7\hat{i}-\hat{k})+(-2\hat{i}+3\hat{j}+5\hat{k})}{\lambda +1}=\hat{i}+2\hat{j}+3\hat{k}$
$\Rightarrow (7\lambda -2)\hat{i}+3\hat{j}+(5-\lambda )\hat{k}=(\lambda +1)\hat{i}+2(\lambda +1)\hat{j}+3(\lambda +1)\hat{k}$
On equating the coefficient of $\hat{i}$, we get
$7\lambda -2=\lambda +1\Rightarrow \lambda =\frac{1}{2}$
Hence, required ratio $=\lambda :1=1:2$