Let \( \hat{u}=u_{1} \hat{i}+u_{2} \hat{j}+u_{3} \hat{k} \) be a unit vector in \( R^{3} \) and
P \( \hat{\omega}=\frac{1}{\sqrt{6}}(\hat{i}+\hat{j}+2 \hat{k}) \). Given that there exists a vector \( \vec{v} \) in
W)
\( R^{3} \) 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?
[JEE Advanced-2016]
(a) There is exactly one choice for such \( \overrightarrow{\mathrm{V}} \)
(b) There are infinitely many choices for such \( \overrightarrow{\mathrm{v}} \)
(c) If lies in the \( x y \)-plane then \( \left|\mathrm{u}_{1}\right|=\left|\mathrm{u}_{2}\right| \)
(d) If lies in the \( x z \)-plane then \( 2\left|\mathrm{u}_{1}\right|=\left|\mathrm{u}_{3}\right| \)
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