Find the shortest distance between the lines given below:

and

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, -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