3D Geometry · Intersecting lines and distances

Let the lines

L_1 : \vec r = \hat i + 2\hat j + 3\hat k + \lambda\left(2\hat i + 3\hat j + 4\hat k\right)

\lambda \in \mathbb{R}

and

L_2 : \vec r = \left(4\hat i + \hat j\right) + \mu\left(5\hat i + 2\hat j + \hat k\right)

\mu \in \mathbb{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\lvert\overrightarrow{PR}\right\rvert = \sqrt{29}

and

\left\lvert\overrightarrow{PQ}\right\rvert = \sqrt{\dfrac{47}{3}}

. If the point

P

lies in the first octant, then

27(QR)^2

is equal to

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