Mathematics - Vector Question with Solution | TestHub

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 $(\overrightarrow{OP}\times \overrightarrow{OQ})\cdot \overrightarrow{OR}=0$ and the point $(\alpha ,\beta ,2)$ lies on the plane $3x+3y-z+l=0$, then the value of $l$ is ____