Given : Cartesian equations of lines

To Find : distance d

Formulae :

1. Equation of line :

Equation of line passing through point A (a₁, a₂, a₃) and having direction ratios (b₁, b₂, b₃) 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.