The unit vector which is perpendicular to \(\hat{i}+\hat{j}+\hat{k}\) and \(\hat{i}+\hat{k}\) is

\(\hat{i}\) \(\hat{i}-\hat{k}\) \(\frac{\hat{i}-\hat{k}}{\sqrt{2}}\) \(\frac{\hat{i}+\hat{k}}{\sqrt{2}}\)

\(\frac{\hat{i}-\hat{k}}{\sqrt{2}}\)

Note that \(\vec{a}\times \vec{b}\)= \(\hat{i}-\hat{k}\) and thus after normalising required vector is \(\frac{\hat{i}-\hat{k}}{\sqrt{2}}\)