Let the lines L_1: r = i +2 j +3 k +λ(2 i +3 j +4 k ), λ ∈ R and L_2: r =(4 i + j )+μ(5 i +2 j + k ), μ ∈ R, intersect at the point R. Let P and Q be…

Let the lines $\displaystyle \mathrm{L}_1: \vec{r}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}+\lambda(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}), \lambda \in \mathbb{R}$ and $\displaystyle \mathrm{L}_2: \vec{r}=(4 \hat{\mathrm{i}}+\hat{\mathrm{j}})+\mu(5 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}), \mu \in \mathbb{R}$, intersect at the point R . Let P and Q be the points lying on lines $\displaystyle \mathrm{L}_1$ and $\displaystyle \mathrm{L}_2$, respectively, such that $\displaystyle |\overrightarrow{\mathrm{PR}}|=\sqrt{29}$ and $\displaystyle |\overrightarrow{\mathrm{PQ}}|=\sqrt{\frac{47}{3}}$. If the point P lies in the first octant, then $\displaystyle 27(\mathrm{QR})^2$ is equal to