A vector coplanar with vectors hat i +hat j and hat j + hat k and par…

Any vector coplanar with vectors $\hat i +\hat j$ and $\hat j + \hat k$ is

$\vec a=x(\hat i +\hat j)+y(\hat j + \hat k)$

or, $\vec a=x\hat i+(x+y)\hat j+y\hat k$

It is given that $\vec a$ is parallel to $2\hat i-2\hat j-4\hat k$

$∴\vec a=λ(2\hat i-2\hat j-4\hat k)$ for some scalar λ

$⇒\{x\hat i+(x+y)\hat j+y\hat k\}=λ(2\hat i-2\hat j-4\hat k)$

$⇒x=2λ,x+y=-2λ$ and $y=-4λ$

$⇒x=2λ$ and $y=-4λ$

$∴\vec a=2λ(\hat i-\hat j-2\hat k)$, where $λ∈R$

Hence, option (2) is true.