Question

Let $L_1$ and $L_2$ denote the lines
$\vec{r}=\hat{i}+\lambda \left(-\hat{i}+2\hat{j}+2\hat{k}\right),\lambda \in \mathbb{R}$ and
$\vec{r}=\mu \left(2\hat{i}-\hat{j}+2\hat{k}\right),\mu \in \mathbb{R}$
respectively. If $L_3$ is a line which is perpendicular to both $L_1$ and $L_2$ and cuts both of them, then which of the following options describe(s) $L_3$?

• A. $\vec{r}=\frac{2}{9}\left(4\hat{i}+\hat{j}+\hat{k}\right)+t\left(2\hat{i}+2\hat{j}-\hat{k}\right),t\in \mathbb{R}$
• B. $\vec{r}=\frac{2}{9}\left(2\hat{i}-\hat{j}+\hat{2}k\right)+t\left(2\hat{i}+2\hat{j}-\hat{k}\right),t\in \mathbb{R}$
• C. $\vec{r}=\frac{1}{3}\left(2\hat{i}+\hat{k}\right)+t\left(2\hat{i}+2\hat{j}-\hat{k}\right),t\in \mathbb{R}$
• D. $\vec{r}=t\left(2\hat{i}+\hat{j}-\hat{k}\right),t\in \mathbb{R}$

Answer 1

Answer: (A), (B), (C)

$\because$ $L_3$ is perpendicular to both $L_1$ and $L_2$.
Then a vector along $L_3$ will be,
$\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ -1& 2& 2\\ 2& -1& 2\end{array}\right|=3\left(2\hat{i}+2\hat{j}-\hat{k}\right)$
Consider a point on $L_{1}asP\left(-\lambda +1,2\lambda ,2\mu -2\lambda \right)$ and a point on $L_2$ as $Q\left(2\mu ,-\mu ,2\mu \right)$
DR's of $L_3<2\mu +\lambda -1,-\mu -2\lambda ,2\mu -2\lambda >$
Here $\frac{2\mu +\lambda -1}{2}=\frac{-\mu -2\lambda }{2}=\frac{2\mu -2\lambda }{-1}$
$\Rightarrow \lambda =\frac{1}{9}$ and $\mu =\frac{2}{9}$
$P\left(\frac{8}{9},\frac{2}{9},\frac{2}{9}\right)$ and $Q\left(\frac{4}{9},-\frac{2}{9},\frac{4}{9}\right);$ mid-point of $PQ$ is $R\left(\frac{2}{3},0,\frac{1}{3}\right)$
Equation of $L_3:\vec{r}=\vec{a}+\lambda \left(2\hat{i}+2\hat{j}-\hat{k}\right),$ here $\bar{a}$ is the position vector of any point on $L_3$. Possible vector
of $\bar{a}$ are $\left(\frac{8}{9}\hat{i}+\frac{2}{9}\hat{j}+\frac{2}{9}\hat{k}\right)or\left(\frac{4}{9}\hat{i}-\frac{2}{9}\hat{j}+\frac{4}{9}\hat{k}\right)or\left(\frac{2}{3}\hat{i}+\frac{1}{3}\hat{k}\right)$