Answer :

__Given__**:** Cartesian equations of lines

__To Find__**:** distance d

__Formulae__**:**

**1. Equation of line :**

Equation of line passing through point A (a_{1}, a_{2}, a_{3}) and having direction ratios (b_{1}, b_{2}, b_{3}) is

Where,

And

**2. Cross Product :**

If are two vectors

then,

**3. Dot Product :**

If are two vectors

then,

**4. Shortest distance between two lines :**

The shortest distance between the skew lines and

is given by,

__Answer__**:**

Given Cartesian equations of lines

Line L1 is passing through point (1, -2, 3) and has direction ratios (-1, 1, -2)

Therefore, vector equation of line L1 is

And

Line L2 is passing through point (1, -1, -1) and has direction ratios (2, 2, -2)

Therefore, vector equation of line L2 is

Now, to calculate distance between the lines,

Here,

Therefore,

Now,

= 0 - 6 + 16

= 10

Therefore, the shortest distance between the given lines is

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