Let L _ 1 and L _ 2 denotes the lines → r = ^ i + λ(- ^ i + 2 ^ j +...

{L}_{1}: \overset{\to}{r} = \overset{^}{i} + \lambda\left(- \overset{^}{i} + 2\overset{^}{j} + 2\overset{^}{k}\right), \lambda \in R

Let a general point on line ${L}_{1}$ be $A\left(1 - \lambda, 2\lambda, 2\lambda\right)$

Also {L}_{2}: \overset{\to}{r} = \mu\left(2\overset{^}{i} - \overset{^}{j} + 2\overset{^}{k}\right)
let a general point on line ${L}_{2}$ be $B\left(2\mu, - \mu, 2\mu\right)$
Then \overset{\to}{AB} = \left(2\mu + \lambda - 1\right)\overset{^}{i} + \left(- \mu - 2\lambda\right)\overset{^}{j} + \left(2\mu - 2\lambda\right)\overset{^}{k}
Now, $\overset{\to}{AB}$ is perpendicular to both ${L}_{1}$ and ${L}_{2}$
Hence, $( a )$ \overset{\to}{AB} ⋅ \left(- \overset{^}{i} + 2\overset{^}{j} + 2\overset{^}{k}\right) = 0 ( If $\overset{\to}{a}$ and $\overset{\to}{b}$ are perpendicular then $\overset{\to}{a} ⋅ \overset{\to}{b} = 0$ )
& $( b )$ \overset{\to}{AB} ⋅ \left(2\overset{^}{i} - \overset{^}{j} + 2\overset{^}{k}\right) = 0

Using
$( a )$ $\overset{\to}{AB} ⋅ \left(- i + 2j + 2k\right) = 0$
\left(- 1\right)\left(2\mu + \lambda - 1\right) + \left(- \mu - 2\lambda\right)\left(2\right) + \left(2\mu - 2\lambda\right)2 = 0$<br />$- 2\mu - \lambda + 1 - 2\mu - 4\mu + 4\mu - 4\lambda = 0$<br />$9\lambda = 1$<br />$\lambda = \dfrac{1}{9} \ldots \left(i\right)$<br />From$( b )$we have$\overset{\to}{AB} ⋅ \left(2\overset{^}{i} - \overset{^}{j} + 2\overset{^}{k}\right) = 0$<br />$⇒ \left(2\mu + \lambda - 1\right)\left(2\right) + \left(- \mu - 2\lambda\right)\left(- 1\right) + \left(2\mu - 2\lambda\right)\left(2\right) = 0$<br />$⇒ 4\mu + 2\lambda - 2 + \mu + 2\lambda + 4\mu - 4\lambda = 0$<br />$⇒ 9\mu = 2$<br />$⇒ \mu = \dfrac{2}{9} \ldots \left(ii\right)$<br />Using value of$\lambda$and$\mu ⇒ A\left(\dfrac{8}{9}, \dfrac{2}{9}, \dfrac{2}{9}\right)$and$B\left(\dfrac{4}{9}, - \dfrac{2}{9}, \dfrac{4}{9}\right)$<br />$\overset{\to}{AB} = - \dfrac{2}{9}\left(2\overset{^}{i} + 2\overset{^}{j} - \overset{^}{k}\right)$<br />Now, using point$A$and$\overset{\to}{AB}$<br />Equation of line$= \left(\dfrac{8}{9}\overset{^}{i} + \dfrac{2}{9}\overset{^}{j} + \dfrac{2}{9}\overset{^}{k}\right) + \gamma\left(- \dfrac{2}{9}\left(2\overset{^}{i} + 2\overset{^}{j} - \overset{^}{k}\right)\right)$<br />Option$( B )$:<br />$\overset{\to}{r} = \dfrac{2}{9}\left(4\overset{^}{i} + \overset{^}{j} + \overset{^}{k}\right) + t\left(2\overset{^}{i} + 2\overset{^}{j} - \overset{^}{k}\right), t \in R$<br />Using point$B$and$\overset{\to}{AB}$<br />Option$( C )$:<br />Equation of line$\overset{\to}{r} = \dfrac{2}{9}\left(2\overset{^}{i} - \overset{^}{j} + 2\overset{^}{k}\right) + t\left(2\overset{^}{i} + 2\overset{^}{j} - \overset{^}{k}\right), t \in R$<br />Using midpoint of$A$and$B$, \left(\dfrac{2}{3}, 0, \dfrac{1}{3}\right)$
Option $( A )$ :
\overset{\to}{r} = \dfrac{1}{3}\left(2\overset{^}{i} + \overset{^}{k}\right) + t\left(2\overset{^}{i} + 2\overset{^}{j} - \overset{^}{k}\right), t \in R