Question

Find the vector equation of a line passing through the point (2, 3, 2) and parallel to the line

r = (-2i + 3j) + λ(2i - 3j + 6k). Also, find the distance between these lines.

HINT: The given line is

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

Answer 1

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

\(\vec r=\bar a_1+\lambda \bar{b}\)

Where,

4. Shortest distance between two parallel lines :

The shortest distance between the parallel lines \(\vec r=\bar a_1+\lambda \bar{b}\) and 

\(\vec r=\bar a_2+\lambda \bar {b}\) is given by,

As the required line is parallel to the line

Therefore, the vector parallel to the required line is

Therefore, equation of line passing through A and parallel to \(\vec b\) is 

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

Therefore, the shortest distance between the given lines is