# Show that the lines  and  intersect.Also, find their point of intersection.

Given equations :  To Find : d

Formula :

1. Cross Product :

If are two vectors  then, 2. Dot Product :

If are two vectors  then, 3. Shortest distance between two lines :

The shortest distance between the skew lines and is given by, For given lines,  Here,    Therefore,        Now,  = - 15 – 18 + 33

= 0

Therefore, the shortest distance between the given lines is   As d = 0

Hence, the given lines not intersect each other.

Now, to find point of intersection, let us convert given vector equations into Cartesian equations.

For that substituting in given equations,      General point on L1 is

x1 = 2λ+1 , y1 = 3λ+2 , z1 = 4λ+3

let, P(x1, y1, z1) be point of intersection of two given lines.

Therefore, point P satisfies equation of line L2.  4λ – 6 = 15λ + 5

11λ = -11

λ = -1

Therefore, x1 = 2(-1)+1 , y1 = 3(-1)+2 , z1 = 4(-1)+3

x1 = -1 , y1 = -1 , z1 = -1

Hence point of intersection of given lines is (-1, -1, -1).

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