Find the vector equation of a line passing through the point (2, 3, 2) and parallel to the line Also, find the distance between these lines.

HINT: The given line is

The required line is

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

Given : point A ≡ (2, 3, 2)

Equation of line :

To Find : i) equation of line

ii) distance d

Formulae :

1. Equation of line :

Equation of line passing through point A (a₁, a₂, a₃) and parallel to vector is given by

Where,

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 :

As the required line is parallel to the line

Therefore, the vector parallel to the required line is

Given point A ≡ (2, 3, 2)

Therefore, equation of line passing through A and parallel to is

Now, to calculate distance between above line and given line,

Here,

= 7

Therefore, the shortest distance between the given lines is