Let u =u_ 1 i +u_ 2 j +u_ 3 k be a unit vector in R 3 and w = 1 6 ( i + j +2 k ) . Given…

Let \(\hat{u}=u_{1} \hat{i}+u_{2} \hat{j}+u_{3} \hat{k}\) be a unit vector in R³ and \(\hat{w}=\frac{1}{\sqrt{6}}(\hat{i}+\hat{j}+2 \hat{k})\). Given that there exists a vector in R³ such that \(|\hat{u} \times \vec{v}|=1\) and \(\hat{w} \cdot(\hat{u} \times \vec{v})=1\). Which of the following statement(s) is(are) correct?