Let the position vectors of points 'A' and 'B' be i + j + k and 2 i + j +3 k, respectively. A point ' P ' divides the line segment AB internally in…

Let the position vectors of points 'A' and 'B' be $\displaystyle \hat{i}+\hat{j}+\hat{k}$ and $\displaystyle 2 \hat{i}+\hat{j}+3 \hat{k}$, respectively. A point ' P ' divides the line segment AB internally in the ratio $\displaystyle \lambda: 1(\lambda>0)$. If O is the origin and $\displaystyle \overrightarrow{\mathrm{OB}} \cdot \overrightarrow{\mathrm{OP}}-3|\overrightarrow{\mathrm{OA}} \times \overrightarrow{\mathrm{OP}}|^2=6$, then $\displaystyle \lambda$ is equal to \_\_\_\_.