Let → a = 2 ^ i - ^ j + 2 ^ k and → b = ^ i + 2 ^ j - ^ k . Let a...

$\overset{\to}{v}, \overset{\to}{a}, \overset{\to}{b}$ are coplanar

$\overset{\to}{v} = \lambda\overset{\to}{a} + \mu\overset{\to}{b}$
\overset{\to}{v} = \left(2\lambda + \mu\right)\overset{^}{i} + \left(2\mu - \lambda\right)\overset{^}{j} + \left(2\lambda - \mu\right)\overset{^}{k}

given \overset{\to}{v} ⊥\left(3\overset{^}{i} + 2\overset{^}{j} - \overset{^}{k}\right)

\overset{\to}{v} \cdot \left(3\overset{^}{i} + 2\overset{^}{j} - \overset{^}{k}\right) = 0

⇒ ( \left(2\lambda + \mu\right)\overset{^}{i} + \left(2\mu - \lambda\right)\overset{^}{j} + \left(2\lambda - \mu\right)\overset{^}{k} ) . ( 3\overset{^}{i} + 2\overset{^}{j} - \overset{^}{k} ) = 0

$⇒ 3 ( 2\lambda + \mu ) + 2 ( 2\mu - \lambda ) - ( 2\lambda - \mu ) = 0$

$⇒ \lambda = - 4\mu . . . ( i )$

and also \overset{\to}{v} \cdot \overset{^}{a} = 19 ⇒ 9\lambda - 2\mu = 57 . . . \left(ii\right)

solving $( i ) and ( ii )$ we get

$\lambda = 6, \mu = - \dfrac{3}{2}$
⇒ 2\overset{\to}{v} = 21\overset{^}{i} - 18\overset{^}{j} + 27\overset{^}{k}
$⇒ | 2\overset{\to}{v}{|}^{2} = 441 + 324 + 729$
$= 1494$