Let $\hat { u } = u _ { 1 } \hat { i } + u _ { 2 } \hat { j } + u _ { 3 } \hat { k }$ be a unit vector in $\mathbb { R } ^ { 3 }$ and $\hat { w } = \frac { 1 } { \sqrt { 6 } } ( \hat { i } + \hat { j } + 2 \hat { k } )$. Given that there exists a vector $\vec { v }$ in $\mathbb { R } ^ { 3 }$ 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? (A) There is exactly one choice for such $\vec { v }$ (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 $2 \left| u _ { 1 } \right| = \left| u _ { 3 } \right|$
Let $\hat { u } = u _ { 1 } \hat { i } + u _ { 2 } \hat { j } + u _ { 3 } \hat { k }$ be a unit vector in $\mathbb { R } ^ { 3 }$ and $\hat { w } = \frac { 1 } { \sqrt { 6 } } ( \hat { i } + \hat { j } + 2 \hat { k } )$. Given that there exists a vector $\vec { v }$ in $\mathbb { R } ^ { 3 }$ 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?\\
(A) There is exactly one choice for such $\vec { v }$\\
(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 $2 \left| u _ { 1 } \right| = \left| u _ { 3 } \right|$