Find the shortest distance between lines ` -> r=6 hat i+2 hat j+ hat k+lambda( hat i-2 hat j+2 hat k)`and ` -> r=-4 hat i- hat k+mu(3 hat i-2 hat j-2 hat k)`.

Explanation

To find the shortest distance between the two given lines, we can use the formula for the shortest distance \(d\) between two skew lines defined by their vector equations:

\[ d = \frac{| \mathbf{b_1} \times \mathbf{b_2} \cdot (\mathbf{a_2} - \mathbf{a_1}) |}{|\mathbf{b_1} \times \mathbf{b_2}|} \]

Where:
- \(\mathbf{a_1}\) and \(\mathbf{a_2}\) are position vectors of points on the lines,
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