The vector (s) which is (are) coplanar with vectors $\hat i+\hat j+2\hat k$ and $\hat i +2\hat j+\hat k$, and perpendicular to the vector $\hat i+\hat j+\hat k$, is/are

$\hat j-\hat k$ and $-\hat j+\hat k$

Required vector are parallel to the vector $\vec r$ given by

$\vec r = \vec a× (\vec b× \vec c)$, where $\vec a =\hat i +\hat j+\hat k, \vec b = \hat i +\hat j+2\hat k$ and $\vec c=\hat i+2\hat j+\hat k$

$⇒\vec r =(\vec a.\vec c)\vec b-(\vec c.\vec b)\vec c$

$⇒\vec r =4(\hat i+\hat j+2\hat k)-4(\hat i+2\hat j+\hat k)=4(-\hat j+\hat k)$

Hence, required vectors are parallel to the vector $±(\hat j-\hat k)$.