Question

Show that the lines $\overrightarrow{r}=(-3\hat{i}+\hat{j}+5\hat{k})+\lambda (-3\hat{i}+\hat{j}+5\hat{k})\text{and}\overrightarrow{r}=(-\hat{i}+2\hat{j}+5\hat{k})+\mu (-\hat{i}+2\hat{j}+5\hat{k})$ are coplanar. Also,find the equation of the plane containing these lines.

Answer 1

$\text{Here}\overrightarrow{a_1}=-3\hat{i}+\hat{j}+5\hat{k},\overrightarrow{b_1}=-3\hat{i}+\hat{j}+5\hat{k},\overrightarrow{a_2}=-\hat{i}+2\hat{j}+5\hat{k},\overrightarrow{b_2}=-\hat{i}+2\hat{j}+5\hat{k}$
$\text{Now}(\overrightarrow{a_2}-\overrightarrow{a_1}).(\overrightarrow{b_1}\times \overrightarrow{b_2})=\left|\begin{array}{ccc}2& 1& 0\\ -3& 1& 5\\ -1& 2& 5\end{array}\right|=2(-5)-1(-15+5)=-10+10=0$
$\therefore$ Given lines are coplanar.
Perpendicular vector to the plane $\overrightarrow{b_1}\times \overrightarrow{b_2}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ -3& 1& 5\\ -1& 2& 5\end{array}\right|=-5\hat{i}+10\hat{j}-5\hat{k}\text{or}\hat{i}-2\hat{j}+\hat{k}$
$\therefore$ Eq.of plane:$\overrightarrow{r}.(\hat{i}-2\hat{j}+\hat{k})=(\hat{i}-2\hat{j}+\hat{k}).(-3\hat{i}+\hat{j}+5\hat{k})\Rightarrow x-2y+z=0$