Prove the following


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

  1. do not intersect for any values of $l$ and $m$

  2. intersect for all values of $l$ and $m$

  3. intersect when $l=2$ and $m=\frac{1}{2}$

  4. intersect when $l=1$ and $m=2$

Correct Option: 1


$L_{1} \equiv \vec{r}=(\hat{i}-\hat{j})+\ell(2 \hat{i}+k)$

$L_{2} \equiv \vec{r}=(2 \hat{i}-\hat{j})+m(\hat{i}+\hat{j}-\hat{k})$

Equating coeff. of $\hat{i}, \hat{j}$ and $\hat{k}$ of $L_{1}$ and $L_{2}$

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

$-1=-1+m \Rightarrow m=0$ 

$\ell=-m$ ...(iii)

$\Rightarrow m=\ell=0$ which is not satisfy eqn. (i) hence lines do not intersect for any value of $\ell$ and $m$.

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