For real numbers \( \alpha, \beta, \gamma, \delta \) and \( \mu \), consider the matrix
.
Suppose that \( MM^{T} = I \), where \( M^{T} \) is the transpose of the matrix \( M \), and \( I \) is the \( 3 \times 3 \) identity matrix.
Let $\vec{u} = \alpha\hat{i} + \frac{1}{\sqrt{3}}\hat{j} + \gamma\hat{k}$, $\vec{v} = \frac{1}{\sqrt{2}}\hat{i} + \beta\hat{j} + \delta\hat{k}$ and $\vec{w} = -\frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{3}}\hat{j} + \mu\hat{k}$.
Match each entry in List-I to the correct entry in List-II and choose the correct option.
