Question

A line passes through the point with position vector $3\hat{i}$ and is parallel to the vector $-\hat{i}+\hat{j}+\hat{k}$ and another line passes through the point with position vector $\hat{i}+\hat{j}$ and is parallel to the vector $\hat{i}+\hat{k}$ then they intersect at the point with position vector

• A. $\hat{i}-2\hat{j}+\hat{k}$
• B. $\hat{i}-2\hat{j}-\hat{k}$
• C. $2\hat{i}+\hat{j}+\hat{k}$
• D. $\hat{i}+2\hat{j}+\hat{k}$

Answer 1

The correct option is C $2\hat{i}+\hat{j}+\hat{k}$

The first line is $\overrightarrow{r}=3\hat{i}+s(-\hat{i}+\hat{j}+\hat{k})$ ..........$\left(1\right)$
The second line is $\overrightarrow{r}=\hat{i}+\hat{j}+t(\hat{i}+\hat{k})\left(2\right)$ ................$\left(2\right)$
$\left(1\right),\left(2\right)$ intersect for
$\left(1\right)\Rightarrow \overrightarrow{r}=(3-s)\hat{i}+s\hat{j}+s\hat{k}$
$\left(2\right)\Rightarrow \overrightarrow{r}=(1+t)\hat{i}+\hat{j}+\hat{k}$
$\Rightarrow 3-s=1+t,s=1,s=t$
$\Rightarrow s=t=1$
$\Rightarrow 3-s=1+t\Rightarrow 3-1=1+1$verified
$\therefore$ the point is $(3-1)\hat{i}+1\hat{j}+1\hat{k}=2\hat{i}+\hat{j}+\hat{k}$