The unit vector which is orthogonal to the vector $3\hat i+2\hat j+6\hat k$ and is coplanar with vectors $2\hat i+\hat j+\hat k$ and $\hat i-\hat j+\hat k$, is

$\frac{1}{\sqrt{10}}(3\hat j-\hat k)$

Let $\vec a=3\hat i+2\hat j+6\hat k, \vec b=2\hat i+\hat j+\hat k$ and $\vec c=\hat i-\hat j+\hat k$.

Then, required unit vectors are given by

Now,

$\vec α=±\frac{\vec a×(\vec b×\vec c)}{|\vec a×(\vec b×\vec c)|}$

Now,

$\vec a×(\vec b×\vec c)=(\vec a.\vec c)\vec b-(\vec a.\vec b)\vec c$

$⇒\vec a×(\vec b×\vec c)=7(2\hat i+\hat j+\hat k)-14(\hat i-\hat j+\hat k)$

$⇒\vec a×(\vec b×\vec c)=21\hat j-7\hat k$

$∴|\vec a×(\vec b×\vec c)|=\sqrt{21^2+7^2}=7\sqrt{10}$

Hence, required unit vectors are

$∴\vec α=±\frac{21\hat j-7\hat k}{7\sqrt{10}}=±\frac{1}{\sqrt{10}}(3\hat j-\hat k)$