Let $\vec u=u_1\hat i+u_2\hat j+u_3\hat k$ be a unit vector in $R^3$ and $\vec w=\frac{1}{\sqrt{6}}(\hat i+\hat j+2\hat k)$. Given that there exists a vector $\vec v$ in $R^3$ such that $|\vec u×\vec v|=1$ and $\vec w(\vec u×\vec v)=1$ which of the following statements is correct?

There are infinitely many choices for such $\vec v$.

Clearly, $\vec w$ is a unit vector such that $|\vec u×\vec v|=1$ and $\vec w(\vec u×\vec v)=1$. Now,

$\vec w(\vec u×\vec v)=1$

$⇒\vec w=\vec u×\vec v$

$⇒|\vec w|=|\vec u×\vec v|$

$⇒1=|\vec u||\vec v|\sin θ$, where θ is the angle between $\vec u$ and $\vec v$

$⇒\vec v\sin θ=1$

Clearly, P can take infinitely many positions on the line at a unit distance from OA. Consequently, $\vec{OP} =\vec v$ has infinitely many choices.