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.

$$M = \begin{bmatrix} \alpha & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \beta & \frac{1}{\sqrt{3}} \\ \gamma & \delta & \mu \end{bmatrix}$$

$$\vec{u} = \alpha\hat{i} + \frac{1}{\sqrt{3}}\hat{j} + \gamma\hat{k}$$

$$\vec{v} = \frac{1}{\sqrt{2}}\hat{i} + \beta\hat{j} + \delta\hat{k}$$

$$\vec{w} = -\frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{3}}\hat{j} + \mu\hat{k}$$

$$\alpha^2 + \frac{1}{3} + \gamma^2 = 1 \quad \therefore \alpha^2 + \gamma^2 = \frac{2}{3} \quad \dots(1)$$

$$\frac{1}{2} + \beta^2 + \delta^2 = 1 \quad \beta^2 + \delta^2 = \frac{1}{2} \quad \dots(2)$$

$$\frac{1}{2} + \frac{1}{3} + \mu^2 = 1 \quad \mu^2 = \frac{1}{6} \quad \dots(3)$$

$$\gamma^2 + \delta^2 + \mu^2 = 1$$

$$\gamma^2 + \delta^2 = 1 - \frac{1}{6} = \frac{5}{6}$$

For Q:

$$x\vec{u} + y\vec{v} + z\vec{w} = \hat{j}$$

$$x\vec{u} \cdot \vec{u} + y\vec{v} \cdot \vec{u} + z\vec{w} \cdot \vec{u} = \vec{u} \cdot \hat{j}$$

$$x = \frac{1}{\sqrt{3}}$$

For R: 

$$\vec{u}, \vec{v}, \vec{w}$$ are mutually perpendicular unit vectors.

$$\vert{}\vec{u} \cdot (\vec{v} \times \vec{w})\vert{}$$

$$\begin{bmatrix} \vec{u} & \vec{v} & \vec{w} \end{bmatrix} = 1$$

For S: 

$$\vec{u} \times (\vec{v} \times \vec{w})$$

$$(\vec{u} \cdot \vec{w})\vec{v} - (\vec{u} \cdot \vec{v})\vec{w}$$

$$= 0 - 0 = 0$$