Find the shortest distance between the lines given below:
and
Given : Cartesian equations of lines
To Find : 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 (-1, 1, -2)
Therefore, vector equation of line L1 is
And
Line L2 is passing through point (1, -1, -1) and has direction ratios (2, 2, -2)
Therefore, vector equation of line L2 is
Now, to calculate distance between the lines,
Here,
Therefore,
Now,
= 0 - 6 + 16
= 10
Therefore, the shortest distance between the given lines is