Let $\vec{OP} = \frac{\alpha - 1}{\alpha}\hat{i} + \hat{j} + \hat{k}$, $\vec{OQ} = \hat{i} + \frac{\beta - 1}{\beta}\hat{j} + \hat{k}$ and $\overline{OR} = \hat{i} + \hat{j} + \frac{1}{2}\hat{k}$ be three vectors, where $a, b \in \mathbb{R} - \{0\}$ and O denotes the origin. If $(\vec{OP} \times \vec{OQ}) \cdot \vec{OR} = 0$ and the point $(\alpha, \beta, 2)$ lies on the plane $3x + 3y - z + l = 0$ then the value of $l$ is ________.

$\vec{OP} = \dfrac{\alpha-1}{\alpha}\hat{i} + \hat{j} + \hat{k},\quad \vec{OQ} = \hat{i} + \dfrac{\beta-1}{\beta}\hat{j} + \hat{k}$ $\vec{OR} = \hat{i} + \hat{j} + \dfrac{1}{2}\hat{k},\quad a, b \in \mathbb{R} - \{0\}$ $(\vec{OP} \times \vec{OQ}) \cdot \vec{OR} = 0$ $(\alpha, \beta, 2) \text{ lies on } 3x + 3y - z + l = 0$ $\text{To find: } l$ $(\vec{OP} \times \vec{OQ}) \cdot \vec{OR} = 0 \Rightarrow \vec{OP}, \vec{OQ}, \vec{OR} \text{ coplanar}$ $\Leftrightarrow \det[\vec{OP}\ \vec{OQ}\ \vec{OR}] = 0$ $\begin{vmatrix} \frac{\alpha-1}{\alpha} & 1 & 1 \\ 1 & \frac{\beta-1}{\beta} & 1 \\ 1 & 1 & \tfrac{1}{2} \end{vmatrix} = 0$ $\text{Let } a = 1-\dfrac{1}{\alpha},\quad b = 1-\dfrac{1}{\beta}$ $a\left(\dfrac{b}{2}-1\right) - \left(\dfrac{1}{2}-1\right) + (1-b) = 0$ $\dfrac{a(b-2)}{2} + \dfrac{3}{2} - b = 0$ $a(b-2) + 3 - 2b = 0 \Rightarrow a(b-2) = 2b-3$ $b = 1-\dfrac{1}{\beta} \Rightarrow b-2 = -\dfrac{\beta+1}{\beta},\quad 2b-3 = -\dfrac{\beta+2}{\beta}$ $\left(1-\dfrac{1}{\alpha}\right)(\beta+1) = \beta+2$ $\dfrac{1}{\alpha} = 1 - \dfrac{\beta+2}{\beta+1} = \dfrac{-1}{\beta+1}$ $\alpha = -(\beta+1) \Rightarrow \alpha + \beta = -1$ $3\alpha + 3\beta - 2 + l = 0 \Rightarrow l = 2 - 3(\alpha+\beta)$ $l = 2 - 3(-1) = 5$ $\boxed{l = 5}$