$\text{ Let }\hat{u}=u_1\hat{i}+u_2\hat{j}+u_3\hat{k}\text{ be a unit vector in }{\mathrm{ℝ}}^3\text{ and }\hat{\omega }=\frac{1}{\sqrt{6}}(\hat{i}+\hat{j}+2\overline{k}).\text{ Given that three }$$\text{ exists a vector }\overrightarrow{v}\text{ in }{\mathrm{ℝ}}^3\text{ such that }|\hat{u}\times \overrightarrow{v}|=1\text{ and }\widehat{w.}(\hat{u}\times \overrightarrow{v})=1\text{ which of the following }$$\text{ statement(s) is (are) correct? }$

$\begin{array}{l}\left|u\right|=1,\left|w\right|=1,\left|u\times v\right|=1\Rightarrow \left|v\right|\sin \left(u,v\right)=1\\ \widehat{w.}\hat{u}\times \overrightarrow{v}=1\Rightarrow \left|w\right|\left|u\times v\right|\cos \left(w,u\times v\right)=1\\ \Rightarrow \cos \left(\omega ,u\times v\right)=1\\ hence \hat{w} is perpendicular to both \hat{u} and \overrightarrow{v} \\ \end{array}$

$As it is given there exists a vector \overrightarrow{v} \Rightarrow \hat{w} must be \perp to \hat{u}$

$hence infinitely many such \overrightarrow{v} exists$

$$\begin{gathered} if \hat{u}=u_1\hat{i}+u_2\hat{j} \\ and \hat{u}.\hat{w}=0 \Rightarrow u_1+u_2=0\Rightarrow \left|u_1\right|=\left|u_2\right| \end{gathered}$$

$$\begin{gathered} if \hat{u}=u_1\hat{i}+u_3\hat{k} \\ and \hat{u}.\hat{w}=0 \Rightarrow u_1+2u_3=0\Rightarrow \left|u_1\right|=2\left|u_3\right| \end{gathered}$$