Let $\vec a = 2\hat i - \hat j + \hat k$, $\vec b = \hat i + 2\hat j - \hat k$ and $\vec c = \hat i + \hat j - 2\hat k$ be three vectors. A vector in the plane of $\vec b$ and $\vec c$ whose projection on $\vec a$ is of magnitude $\sqrt{\frac{2}{3}}$ is

$2\hat i+3\hat j-3\hat k$ or $-2\hat i-\hat j+5\hat k$

Let $\vec R$be a vector in the plane of b and c

$⇒\vec R=(\hat i+2\hat j-\hat k)+μ(\hat i+\hat j-2\hat k)$

Its projection on $\vec a=\frac{\vec a.\vec R}{|\vec a|}\frac{1}{\sqrt{6}}[2+2μ-2-μ-1-2μ]=\frac{-(1+μ)}{\sqrt{6}}$

$⇒\frac{-(1+μ)}{\sqrt{6}}=±\sqrt{\frac{2}{3}}⇒-( 1+μ) = ± 2 ⇒ μ = 1, -3$

$⇒R≡2\hat i+3\hat j-3\hat k$ and $-2\hat i-\hat j+5\hat k$

Hence (A) is the correct answer.