Let the lines \({L}_{1}:\vec{r}=\overset{^}{i}+2\overset{^}{j}+3\overset{^}{k}\) \(+\text{ }\lambda \left(2\overset{^}{i…

Let the lines \({L}_{1}:\vec{r}=\overset{^}{i}+2\overset{^}{j}+3\overset{^}{k}\)

\(+\text{ }\lambda \left(2\overset{^}{i}+3\overset{^}{j}+4\overset{^}{k}\right),\lambda \in R\) and

\({L}_{2}:\vec{r}\text{ }=\text{ }\left(4\overset{^}{i}\text{ }+\text{ }\overset{^}{j}\right)\text{ }+\text{ }\mu \left(5\overset{^}{i}\text{ }+\text{ }2\overset{^}{j}+\overset{^}{k}\right),\)

\(\mu \text{ }\in \text{ }R,\) intersect at the point R.

Let P and Q be the points lying on lines

\({L}_{1}\) and \({L}_{2}\), respectively, such that

\(\left|\vec{PR}\right|=\sqrt{29}\) and \(\left|\vec{PQ}\right|=\sqrt{\frac{47}{3}}.\)

If the point P lies in the first octant, then \(27{\left(QR\right)}^{2}\) is equal to