# Solve The Following Questions

Question:

The lines\

$\overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}-\hat{\mathrm{j}})+\ell(2 \hat{\mathrm{i}}+\hat{\mathrm{k}})$ and

$\overrightarrow{\mathrm{r}}=(2 \hat{\mathrm{i}}-\hat{\mathrm{j}})+\mathrm{m}(\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}})$

1. Intersect when $\ell=1$ and $\mathrm{m}=2$

2. Intersect when $\ell=2$ and $m=\frac{1}{2}$

3. Do not intersect for any values of $\ell$ and $\mathrm{m}$

4. Intersect for all values of $\ell$ and $\mathrm{m}$

Correct Option: 3,

Solution:

$\overrightarrow{\mathrm{r}}=\hat{\mathrm{i}}(1+2 \ell)+\hat{\mathrm{j}}(-1)+\hat{\mathrm{k}}(\ell)$

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

For intersection

$1+2 \ell=2+m$ ....(i)

$-1=m-1$  .....(ii)

$\ell=-\mathrm{m}$   .....(iii)

from (ii) $m=0$

from (iii) $\ell=0$

These values of $m$ and $\ell$ do not satisfy equation (1).

Hence the two lines do not intersect for any values of $\ell$ and $\mathrm{m}$.