Write the vector equation of the following lines and hence find the shortest distance between them :

and

Given : Cartesian equations of lines

To Find : i) vector equations of given lines

ii) 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 (2, 3, 4)

Therefore, vector equation of line L1 is

And

Line L2 is passing through point (2, 3, 5) and has direction ratios (3, 4, 5)

Therefore, vector equation of line L2 is

Now, to calculate distance between the lines,

Here,

Therefore,

Now,

= - 2 + 2 - 2

= -2

Therefore, the shortest distance between the given lines is