Let \( \vec{W}=\hat{i}+\hat{j}-2\hat{k} \), and \( \vec{u} \) and \( \vec{v} \) be two vector, such that \( \vec{u}\times\vec{v}=\vec{w} \) and \( \vec{v}\times\vec{w}=\vec{u} \). Let \( \alpha, \beta, \gamma, \) and \( t \) be real numbers such that \( \vec{u}=\alpha\hat{i}+\beta\hat{j}+\gamma\hat{k} \), \( -t\alpha+\beta+\gamma=0 \), \( \alpha-t\beta+\gamma=0 \), and \( \alpha+\beta-t\gamma=0 \).
Match each entry in List-I to the correct entry in List-II and choose the correct option.
