Question

Find the equation of the line passing through the point $\hat{i}+\hat{j}-3\hat{k}$ and perpendicular to the lines $\overrightarrow{r}=\hat{i}+\mathrm{\lambda }\left(2\hat{i}+\hat{j}-3\hat{k}\right)\mathrm{and}\overrightarrow{r}=\left(2\hat{i}+\hat{j}-\hat{k}\right)+\mathrm{\mu }\left(\hat{i}+\hat{j}+\hat{k}\right).$

Answer 1

The required line is perpendicular to the lines parallel to the vectors $\overrightarrow{b_1}=2\hat{i}+\hat{j}-3\hat{k}\mathrm{and}\overrightarrow{b_2}=\hat{i}+\hat{j}+\hat{k}$.

So, the required line is parallel to the vector $\overrightarrow{b}=\overrightarrow{b_1}\times \overrightarrow{b_2}$.

Now,
$$\begin{gathered} \overrightarrow{b}=\overrightarrow{b_1}\times \overrightarrow{b_2} \\ =\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& 1& -3\\ 1& 1& 1\end{array}\right| \\ =4\hat{i}-5\hat{j}+\hat{k} \end{gathered}$$

Equation of the required line passing through the point $\left(\hat{i}+\hat{j}-3\hat{k}\right)$ and parallel to $\left(4\hat{i}-5\hat{j}+\hat{k}\right)$ is $\overrightarrow{r}=\left(\hat{i}+\hat{j}-3\hat{k}\right)+\lambda \left(4\hat{i}-5\hat{j}+\hat{k}\right)$.