Write the vector equation of the following lines and hence find the shortest distance between them :
and 
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 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 (2, 3, 4)
Therefore, vector equation of line L1 is
And
Line L2 is passing through point (2, 3, 5) and has direction ratios (3, 4, 5)
Therefore, vector equation of line L2 is
Now, to calculate distance between the lines,
Here,
Therefore,
Now,
= - 2 + 2 - 2
= -2
Therefore, the shortest distance between the given lines is
