Show that the lines 
and 
intersect and find their point of intersection.
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 (0, 2, -3) and has direction ratios (1, 2, 3)
Therefore, vector equation of line L1 is
And
Line L2 is passing through point (2, 6, 3) and has direction ratios (2, 3, 4)
Therefore, vector equation of line L2 is
Now, to calculate distance between the lines,
Here,
Therefore,
Now,
= - 2 + 8 – 6
= 0
Therefore, the shortest distance between the given lines is
As d = 0
Hence, given lines intersect each other.
Now, general point on L1 is
x1 = λ , y1 = 2λ+2 , z1 = 3λ-3
let, P(x1, y1, z1) be point of intersection of two given lines.
Therefore, point P satisfies equation of line L2.
⇒ 3λ – 6 = 4λ – 8
⇒ λ = 2
Therefore, x1 = 2 , y1 = 2(2)+2 , z1 = 3(2)-3
⇒ x1 = 2 , y1 = 6 , z1 = 3
Hence point of intersection of given lines is (2, 6, 3).