Find the shortest distance between the lines given below:
and
HINT: Change the given equations in vector form.
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 (12, 1, 5) and has direction ratios (-9, 4, 2)
Therefore, vector equation of line L1 is
And
Line L2 is passing through point (23, 10, 23) and has direction ratios (-6, -4, 3)
Therefore, vector equation of line L2 is
Now, to calculate distance between the lines,
Here,
Therefore,
= 65
Now,
= 220 + 135 + 1080
= 1435
Therefore, the shortest distance between the given lines is