Let $L_{1}$ and $L_{2}$ denote the lines $\vec{r} = \hat{i} + λ (- \hat{i} + 2 \hat{j} + 2 \hat{k}), λ \in R$ and $\vec{r} = μ (2 \hat{i} - \hat{j} + 2 \hat{k}), μ \in R$ respectively. If $L_{3}$ is a line which is perpendicular to both $L_{1}$ nd $L_{2}$ and cuts both of them, then which of the following options describe(s) $L_{3}$ ?

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$\vec{r} = \frac{2}{9} (2 \hat{i} - \hat{j} + 2 \hat{k}) + t (2 \hat{i} + 2 \hat{j} - \hat{k}), t \in ℝ$

$\vec{r} = t (2 \hat{i} + 2 \hat{j} - \hat{k}), t \in ℝ$

$\vec{r} = \frac{1}{3} (2 \hat{i} + \hat{k}) + t (2 \hat{i} + 2 \hat{j} - \hat{k}), t \in ℝ$

$\vec{r} = \frac{2}{9} (4 \hat{i} + \hat{j} + \hat{k}) + t (2 \hat{i} + 2 \hat{j} - \hat{k}), t \in ℝ$

answer is A, C, D.

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Detailed Solution

$\begin{array}{l} The given lines are \\ L_{1} \equiv \vec{r} = \hat{i} + λ (- \hat{i} + 2 \hat{j} + 2 \hat{k}), λ \in R and L_{2} \equiv \vec{r} = μ (2 \hat{i} - \hat{j} + 2 \hat{k}), μ \in R \\ {The line L}_{3} is perpendicular to the above lines \\ {The direction ratios of the line L}_{3} can be obtained by finding cross prodcut of two vectors \\ (- \hat{i} + 2 \hat{j} + 2 \hat{k}) and (2 \hat{i} - \hat{j} + 2 \hat{k}), \\ H e n c e t h e d i r e c t i o n r a t i o s o f L_{3} a r e <2, 2, - 1> \end{array}$

$\begin{array}{l} Let P on L_{1} be (1 - λ, 2 λ, 2 λ) \\ \begin{array}{l} Q on be (2 μ, μ, 2 μ) \bar{P Q} ‖ (2, 2, - 1) \\ ∴ \frac{1 - λ - 2 μ}{2} = \frac{2 λ + μ}{2} = \frac{2 λ - 2 μ}{- 1} \\ Solving 3 λ + 3 μ = 1 \dots \dots \dots \dots (1) λ = 1 / 9 \\ 6 λ - 3 μ = 0 ......... (2) μ = 2 / 9 \\ ∴ P = (\frac{8}{0}, \frac{2}{0}, \frac{2}{0}), Q = (\frac{4}{0}, - \frac{2}{0}, \frac{4}{0}) \end{array} \end{array}$