question_answer

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

A)  There are infinitely many choices for such \[\hat{v}\]

B)  If \[\hat{u}\]lies in the xy-plane then \[|{{u}_{1}}|\,=\,|{{u}_{2}}|\]

C)  If \[\hat{u}\] lies in the xz-plane then \[|{{u}_{1}}|\,=2|{{u}_{3}}|\]

D)  All of these

E)  None of these