Let a unit vector which makes an angle of \(60^{\circ}\) with \(2 \hat{i}+2 \hat{j}-\hat{k}\) and an angle of \(45^{\circ}\) with \(\hat{i}-\hat{k}\) be \(\overrightarrow{\mathrm{C}}\). Then \(\overrightarrow{\mathrm{C}}+\left(-\frac{1}{2} \hat{\mathrm{i}}+\frac{1}{3 \sqrt{2}} \hat{\mathrm{j}}-\frac{\sqrt{2}}{3} \hat{\mathrm{k}}\right)\) is :

1. A \(-\frac{\sqrt{2}}{3} \hat{i}+\frac{\sqrt{2}}{3} \hat{j}+\left(\frac{1}{2}+\frac{2 \sqrt{2}}{3}\right) \hat{k}\)
2. B  \(\frac{\sqrt{2}}{3} \hat{\mathrm{i}}+\frac{1}{3 \sqrt{2}} \hat{\mathrm{j}}-\frac{1}{2} \hat{\mathrm{k}}\)
3. C  \(\left(\frac{1}{\sqrt{3}}+\frac{1}{2}\right) \hat{i}+\left(\frac{1}{\sqrt{3}}-\frac{1}{3 \sqrt{2}}\right) \hat{j}+\left(\frac{1}{\sqrt{3}}+\frac{\sqrt{2}}{3}\right) \hat{k}\)
4. D \(\frac{\sqrt{2}}{3} \mathrm{i}-\frac{1}{2} \hat{k}\)