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, -1, 1) and has direction ratios (3, 2, 5)

Therefore, vector equation of line L1 is

And

Line L2 is passing through point (2, 1, -1) and has direction ratios (2, 3, -2)

Therefore, vector equation of line L2 is

Now, to calculate distance between the lines,

Here,

Therefore,

Now,

= - 19 + 32 – 10

= 3

Therefore, the shortest distance between the given lines is

As d ≠ 0

Hence, given lines do not intersect each other.

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