# The Point of intersection of two Lines

### Problem:Find the point of intersection of the line having the position vector equation r1 = [0, 0, 1]  +  t[1, -1, 1] with the line having the position vector equation r2 = [4, 1, 2]  +  s[-6, -4, 0].

A diagram of this is shown on the right. O is the origin.

P is the point of intersection of the two lines.

and   are the position vectors of any point on the respective lines.
The position vectors are shown drawn at the point of intersection of the two lines where =  . Since, at the point of intersection, the two position vectors are identical it follows that;

[0, 0, 1] + t[1, -1, 1] = [4, 1, 2]   + s[-6, -4, 0]

From which t and s can be solved.
Substituting for t in or, s, in will give the required point. Solving for the value of s and t which satisfies;

[0, 0, 1] + t[1, -1, 1] = [4, 1, 2]  + s[-6, -4, 0]

Rearranging;
t[1, -1, 1] - s[-6, -4, 0 ]  =  [4, 1, 2] -  [0, 0, 1]

[ t -  -6s,     -t  -  -4s,       t ]  =     [4, 1, 1]

Equating components;
t +  6s  = 4                         1.
-t  +  4s  = 1                         2.
t  = 1                        3.

Equations 2. and 3. inply;            t  =  1           s  =

Substituting in the equations for = [0, 0, 1] + t [1, -1, 1]                     or,           = [4, 1, 2]+ [-6, -4, 0]    gives;

= [0, 0, 1] + 1 [1, -1, 1]                                        = [4, 1, 2]+ [-6, -4, 0]

= [0  +  1,   0  +  -1,   1  +  1]                                = [4 +  -3,    1  +   -2,    2  +  0]

= [ 1,-1,  2]                                                            = [1,  -1,  2]

The two lines therefore intersect at the point (1,  -1,  2)