Question

Find the weather lines $\overrightarrow{r}=\hat{i}-\hat{j}-\hat{k}+\lambda (2\hat{i}+\hat{j})\&\overrightarrow{r}=2\hat{i}-\hat{j}+m(\hat{i}+\hat{j}-\hat{k})$ intersect or not.

Answer 1

$\begin{array}{c}\overrightarrow{r}=\hat{i}-\hat{j}-\hat{k}+\lambda \left(2\hat{i}+\hat{j}\right)\\ \overrightarrow{r}=2\hat{i}-\hat{j}+m\left(\hat{i}+\hat{j}-\hat{k}\right)\end{array}$

Distance b/w lines

$\begin{array}{c}=\frac{\left|\right(\hat{i}+\hat{k}).\left(2\hat{i}+\hat{j}\right)\times \left(\hat{i}+\hat{j}-\hat{k}\right)|}{\left|\left(2\hat{i}+\hat{j}\right)\times \left(\hat{i}+\hat{j}-\hat{k}\right)\right|}\\ =\frac{\left|\right(\hat{i}+\hat{k}).\left(-\hat{i}+2\hat{j}+\hat{k}\right)|}{\left|\left(-\hat{i}+2\hat{j}+\hat{k}\right)\right|}=0\end{array}$

Hence,Lines intersect