Question

For the given lines $\overrightarrow{r}=-\hat{i}+4\hat{j}+2\hat{k}+\lambda (\hat{i}-2\hat{j}+\hat{k})$ and $\overrightarrow{r}=2\hat{i}+3\hat{j}+\hat{k}+\mu (\hat{i}+3\hat{j}-3\hat{k}),$ which of the following is correct ?

• A. Lines are parallel
• B. Lines are coincident
• C. Lines are coplanar
• D. Lines are skew

Answer 1

The correct option is C Lines are coplanar

Given lines:
$\overrightarrow{r}=-\hat{i}+4\hat{j}+2\hat{k}+\lambda (\hat{i}-2\hat{j}+\hat{k})=\overrightarrow{a}+\lambda \overrightarrow{b}$ and $\overrightarrow{r}=2\hat{i}+3\hat{j}+\hat{k}+\mu (\hat{i}+3\hat{j}-3\hat{k})=\overrightarrow{c}+\mu \overrightarrow{d}$
As $\overrightarrow{b}$ is not parallel to $\overrightarrow{d}$
So, lines are either intersecting or skew.
Now any parametric point on $L_1=(-1+\lambda ,4-2\lambda ,2+\lambda )$ and on $L_2=(2+\mu ,3+3\mu ,1-3\mu )$
For point of intersection : $L_1=L_2$
$\Rightarrow -1+\lambda =2+\mu \cdots \left(i\right)4-2\lambda =3+3\mu \cdots \left(ii\right)2+\lambda =1-3\mu \cdots \left(iii\right)$
Solving $\left(i\right)$ and $\left(ii\right):$ we get $\lambda =2,\mu =-1$
which satisfies $\left(iii\right)$
So, point of intersection is $(1,0,4)$
i.e. lines are coplanar.