Question

Find the vector equation of a line passing through a point $2\hat{i}+\hat{j}-3\hat{k}$ and perpendicular to the vectors $\hat{i}+\hat{j}+\hat{k}$ and $\hat{i}+2\hat{j}-\hat{k}$ ?

Answer 1

Let the position vector of given point be

$\begin{array}{c}\overrightarrow{a}=2\hat{i}+\hat{j}-3\hat{k}\\ \\ \overrightarrow{b}=\hat{i}+\hat{j}+\hat{k}\\ \\ \overrightarrow{c}=\hat{i}+2\hat{j}-\hat{k}\end{array}$

$\overrightarrow{b}\times \overrightarrow{c}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 1& 1\\ 1& 2& -1\end{array}\right|$

$=-3\hat{i}+2\hat{j}+\hat{k}$

Since the required linre is perpendicular to both $\overrightarrow{b}$ and $\overrightarrow{c}$.

therefore,

Required equation of the line

$\overrightarrow{r}=\overrightarrow{a}+\lambda (\overrightarrow{b}\times \overrightarrow{c})$

$=(2\hat{i}+\hat{j}-3\hat{k})+\lambda (-3\hat{i}+2\hat{j}+\hat{k})$