A unit vector perpendicular to both hat i +hat j and hat j+hat k, is

Let $\vec a=\hat i +\hat j$ and $\vec b=\hat j+\hat k$. Then,

$\vec a×\vec b=\begin{vmatrix}\hat i&\hat j&\hat k\\1&1&0\\0&1&1\end{vmatrix}=\hat i-\hat j+\hat k$

∴ Required vectors = $±\frac{\hat i-\hat j+\hat k}{\sqrt{1^2+(-1)^2+1^2}}=±\frac{\hat i-\hat j+\hat k}{\sqrt{3}}$