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 $\overrightarrow{\mathrm{v}}$ in $\mathrm{R}^3$ such that $|\hat{\mathrm{u}} \times \overrightarrow{\mathrm{v}}|=1$ and $\hat{\mathrm{w}} \cdot(\hat{\mathrm{u}} \times \overrightarrow{\mathrm{v}})=1$. Which of the following statement(s) is(are) correct ?
Select ALL correct options:
A
There is exactly one choice for such $\overrightarrow{\mathrm{v}}$
B
There are infinitely many choices for such $\vec{v}$
C
If $\hat{u}$ lies in the $x y$-plane then $\left|u_1\right|=\left|u_2\right|$
D
If $\hat{u}$ lies in the $x z$-plane then $2\left|u_1\right|=\left|u_3\right|$
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