Let $$\hat{u} = u_1\hat{i} + u_2\hat{j} + u_3\hat{k}$$ be a unit vector in $$R^3$$ and $$\hat{\omega} = \frac{1}{\sqrt{6}}(\hat{i} + \hat{j} + 2 \hat{k})$$. Given that there exists a vector $$\overrightarrow{v}$$ in $$R^3$$ such that $$\mid \hat{u} \times \overrightarrow{v} \mid = 1$$ and $$\overrightarrow{\omega} . (\hat{u} \times \overrightarrow{v}) = 1$$. Which of the following statement(s) is(are) correct?