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