The unit vector which is perpendicular to i+j+k and i+k is

Note that

\(\vec{a}\times \vec{b}\)= \(\hat{i}-\hat{k}\)

$\hat p=\frac{(\hat{i}-\hat{k})}{|\hat{i}-\hat{k}|}$

and thus after normalising required vector is \(\frac{\hat{i}-\hat{k}}{\sqrt{2}}\)