JEE Advanced 2024 Mathematics question on Vector Algebra

Question

Let

\overrightarrow{OP} = \frac{\alpha-1}{\alpha}\hat{i} + \hat{j} + \hat{k}

,

\overrightarrow{OQ} = \hat{i} + \frac{\beta-1}{\beta}\hat{j} + \hat{k}

and

\overrightarrow{OR} = \hat{i} + \hat{j} + \frac{1}{2}\hat{k}

be three vectors, where

\alpha, \beta \in \mathbb{R} - \{0\}

and

O

denotes the origin. If

\left( \overrightarrow{OP} \times \overrightarrow{OQ} \right) \cdot \overrightarrow{OR} = 0

and the point

(\alpha, \beta, 2)

lies on the plane

3x + 3y - z + l = 0

, then the value of

l

is _______.