The shortest distance between the lines
\[
\vec r=\frac13\hat i+2\hat j+\frac83\hat k+\lambda(2\hat i-5\hat j+6\hat k)
\]
\[
\vec r=\left(-\frac23\hat i-\frac13\hat k\right)+\mu(\hat j-\hat k), \quad \lambda,\mu\in\mathbb R
\]
is
Show Hint
Shortest distance between two skew lines uses the scalar triple product formula
\(\frac{|(\vec a_1-\vec a_2)\cdot(\vec b_1\times\vec b_2)|}{|\vec b_1\times\vec b_2|}\).