For real numbers $$\alpha,\beta,\gamma,\delta$$ and $$\mu$$, consider the matrix $$M=\begin{bmatrix}\alpha&\tfrac{1}{\sqrt{2}}&-\tfrac{1}{\sqrt{2}}\\[2pt]\tfrac{1}{\sqrt{3}}&\beta&\tfrac{1}{\sqrt{3}}\\[2pt]\gamma&\delta&\mu\end{bmatrix}.$$

Suppose that $$MM^{T}=I$$, where $$M^{T}$$ is the transpose of $$M$$ and $$I$$ is the $$3\times 3$$ identity matrix. Let $$\vec{u}=\alpha\hat{i}+\tfrac{1}{\sqrt{3}}\hat{j}+\gamma\hat{k},\quad \vec{v}=\tfrac{1}{\sqrt{2}}\hat{i}+\beta\hat{j}+\delta\hat{k},\quad \vec{w}=-\tfrac{1}{\sqrt{2}}\hat{i}+\tfrac{1}{\sqrt{3}}\hat{j}+\mu\hat{k}.$$

Match each entry in List-I to the correct entry in List-II and choose the correct option.