Write the vector equation of each of the following lines and hence determine the distance between them :

and

HINT: The given lines are

Now, find the distance between the parallel lines L₁ and L₂.

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 parallel lines :

The shortest distance between the parallel lines and

is given by,

Answer :

Given Cartesian equations of lines

Line L1 is passing through point (1, 2, -4) and has direction ratios (2, 3, 6)

Therefore, vector equation of line L1 is

And

Line L2 is passing through point (3, 3, -5) and has direction ratios (4, 6, 12)

Therefore, vector equation of line L2 is

Now, to calculate distance between the lines,

Here,

As , given lines are parallel to each other.

Therefore,

= 7

Therefore, the shortest distance between the given lines is