Given vectors are
\(\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}} \text { and } \mathbf{c}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}\)
Then, \(\mathbf{a} \times \mathbf{b}=\left|\begin{array}{ccc}\hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ 1 & 1 & 1 \\ 1 & 1 & 2\end{array}\right|\)
\(=\hat{\mathbf{i}}(2-1)-\hat{\mathbf{j}}(2-1)+\hat{\mathbf{k}}(\mathrm{l}-1)=\hat{\mathbf{i}}-\hat{\mathbf{j}}\)
So, unit vector perpendicular to both \(\mathbf{a}\) and \(\mathbf{b}\) is
\(= \pm \frac{\hat{\mathbf{i}}-\hat{\mathbf{j}}}{\sqrt{2}}\)
Now the magnitude of the projection of \(\pm \frac{\hat{\mathbf{i}}-\hat{\mathbf{j}}}{\sqrt{2}}\) on \(\mathbf{c}\) is
\(\begin{aligned}
& =\left| \pm\left(\frac{\hat{\mathbf{i}}-\hat{\mathbf{j}}}{\sqrt{2}}\right) \cdot \frac{(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}})}{\sqrt{4+9+16}}\right| \\
& =\left|\frac{2-3}{\sqrt{2} \sqrt{29}}\right|=\frac{1}{\sqrt{58}}
\end{aligned}\)
Hence, option (c) is correct.