Let \(\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}, \mathbf{b}=\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\mathbf{c}=\hat{\mathbf{i}}+\hat{\mathbf{k}}\). If \(\mathbf{d}\) is a unit vector such that \(\mathbf{a} \cdot \mathbf{d}=0\) and \(\mathbf{b} \cdot(\mathbf{c} \times \mathbf{d})=0\) and \(\mathbf{d}=\)

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