JEE Main 2020 — Three Dimensional Geometry Question with Solution

Lines $\overrightarrow{r}=\left(\hat{i}-\hat{j}\right)+l\left(2\hat{i}+\hat{k}\right)$ and $\overrightarrow{r}=\left(2\hat{i}-\hat{j}\right)+m\left(\hat{i}+\hat{j}-\hat{k}\right)$

Clearly both lines do not have same direction means lines are not parallel.

$\Rightarrow$Lines are skew lines or intersecting lines.

Let lines are skew lines and distance between lines is $d$ if distance between them comes out zero then they will be intersecting lines.

For lines $\overrightarrow{r}=\overrightarrow{a}+\lambda \overrightarrow{p}, \overrightarrow{r}=\overrightarrow{b}+\lambda \overrightarrow{q}$

$d=\frac{\left(\overrightarrow{b}-\overrightarrow{a}\right)·\left(\overrightarrow{p}\times \overrightarrow{q}\right)}{\left|\overrightarrow{p}\times \overrightarrow{q}\right|}$

Applying here

$d=\frac{\left(2\hat{i}-\hat{j}-\left(\hat{i}-\hat{j}\right)\right)·\left(\left(2\hat{i}+\hat{k}\right)\times \left(\hat{i}+\hat{j}-\hat{k}\right)\right)}{\left|\left(2\hat{i}+\hat{k}\right)\times \left(\hat{i}+\hat{j}-\hat{k}\right)\right|}$

$d=\frac{\left(\hat{i}\right)·\left(-\hat{i}+3\hat{j}+2\hat{k}\right)}{\left|-\hat{i}+3\hat{j}+2\hat{k}\right|}=\left|-\frac{1}{\sqrt{14}}\right|=\frac{1}{\sqrt{14}}$

$\Rightarrow$Lines are skew lines means do not intersect.