Question

Show that the lines $\overline{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda (3\hat{i}-\hat{j})$ and $\overline{r}=(4\hat{i}-\hat{j})+\mu (2\hat{i}-\widehat{3k})$ intersect. Find their point of intersection.

Answer 1

Given line are

$\begin{array}{c}\overline{r}=\left(\hat{i}+\hat{j}-\hat{k}\right)+\lambda \left(3\hat{i}-\hat{j}\right)and\\ \overline{r}=\left(4\hat{i}-\hat{j}\right)+\mu \left(2\hat{i}-3\hat{k}\right)\end{array}$

These two line will be intersect if their correspondent coefficient are equal

$\begin{array}{c}\overline{r}=\left(\hat{i}+\hat{j}-\hat{k}\right)+\lambda \left(3\hat{i}-\hat{j}\right)and\\ \overline{r}=\left(4\hat{i}-\hat{j}\right)+\mu \left(2\hat{i}-3\hat{k}\right)\\ \hat{i}\left(1+3\lambda \right)+\hat{j}\left(1-\lambda \right)+\hat{k}\left(-1\right)=\hat{i}\left(4+2\mu \right)+\hat{j}\left(-1\right)+\hat{k}\left(-3\mu \right)\\ so1-\lambda =-1\\ \lambda =2\\ 1+3\lambda =4+2\mu \\ 1+3\left(2\right)-4=2\mu \\ \mu =\frac{3}{2}\\ put\lambda =2\\ \overline{r}=\left(\hat{i}+\hat{j}-\hat{k}\right)+2\left(3\hat{i}-\hat{j}\right)\\ =7\hat{i}-\hat{j}-\hat{k}\\ \end{array}$

Point of intersection is (7 , -1 -1 )