Question

The lines $\overrightarrow{r}=(\hat{i}-\hat{j})+l(2\hat{i}+\hat{k})$ and $\overrightarrow{r}=(2\hat{i}-\hat{j})+m(\hat{i}+\hat{j}-\hat{k})$

• A. do not intersect for any values of $l$ and $m$
• B. intersect when $l=1$ and $m=2$
• C. intersect when $l=2$ and $m=\frac{1}{2}$
• D. intersect for all values of $l$ and $m$

Answer 1

The correct option is A do not intersect for any values of $l$ and $m$

Given:
$\overrightarrow{r}=(\hat{i}-\hat{j})+l(2\hat{i}+\hat{k})$
$\Rightarrow \overrightarrow{r}=\hat{i}(1+2l)+\hat{j}(-1)+\hat{k}\left(l\right)$ and
$\overrightarrow{r}=(2\hat{i}-\hat{j})+m(\hat{i}+\hat{j}-\hat{k})$
$\Rightarrow \overrightarrow{r}=\hat{i}(2+m)+\hat{j}(-1+m)+\hat{k}(-m)$
If these two lines intersect each other, then coefficients of $\hat{i},\hat{j},\hat{k}$ are equal.
$\Rightarrow 1+2l=2+m\cdots \left(i\right)$
$\Rightarrow -1=m-1\Rightarrow m=0$
$\Rightarrow l=-m\Rightarrow l=0$
These values of $m$ and $l$ do not satisfy eq.$\left(i\right)$
Hence, the two given lines do not intersect for any values of $l$ and $m.$