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

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