# How do I find the intersection of two lines in three-dimensional space?

##### 1 Answer

I'll show how to solve this problem using an example using the vector based form of a two straight lines. Bear in mind that there will be one of the following outcomes:

- a single unique point.
- no solution (if the lines do not intersect).
- infinitely many solutions (if the lines coincide).

Suppose we have;

#L_1: \ \ \ \ vec(r_1) = ((5),(2),(-1)) +lamda((1),(-2),(-3))#

#L_2: \ \ \ \ vec(r_2) = ((2),(0),(4)) +mu((1),(2),(-1))#

It is vital that the variable parameters (

If the lines do meet then for some specific values of

#vec(r_1) = vec(r_2)#

In which case we would have:

#((5),(2),(-1)) +lamda((1),(-2),(-3)) = ((2),(0),(4)) +mu((1),(2),(-1))#

# :. \ \ \ \ \ \ ((5+lamda),(2-2lamda),(-1-3lamda)) = ((2+mu),(2mu),(4-mu))#

By comparing the coefficients of

# hat(i) : \ \ \ \ \ \ \ \ \ \ \ 5+lamda = 2+mu " " ....... [1]#

# hat(j) : \ \ \ \ \ \ \ \ \ 2-2lamda = 2mu " " ....... [2] #

# hat(k) : \ \ \ \ -1-3lamda = 4-mu \" " ....... [3] #

Subs

Subs

So we have established that if we choose

We can then substitute

#vec(r_1) = ((5),(2),(-1)) -((1),(-2),(-3))#

# \ \ \ = ((5-1),(2+2),(-1+3))#

# \ \ \ = ((4),(4),(2))#

so in this example, the lines