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

Question

There is exactly one choice for such $\vec{v}$.

There are exactly two choices for such $\vec{v}$.

There are exactly four choices for such $\vec{v}$.

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

Solution

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