# Distance between 2 parallel line in 3-Dimensional Space.

1. Jul 30, 2011

### icystrike

1. The problem statement, all variables and given/known data

What are the steps involved in it?
I have my own way of doing it but i'm just curious to know how it is usually done.

2. Relevant equations

3. The attempt at a solution

Last edited: Jul 30, 2011
2. Jul 30, 2011

### thepatient

Distance formula.

Or, you can find a vector that points from an arbitrary point of one line L1 and ends on an arbituary point of the other line L2, then find the angle between vector of L1 and vector used from L1 to L2, then use trigonometry.

Or you can use the fact that for the dot product of two vectors u, v, then:
u*v = |u||v|cos(theta). From the dot product you can derive the distance formula though.

Let u be a vector from an arbitrary point in L1 to one in L2, and v be the vector of L2, then:
u*v/|v| =|u|cos(theta) = distance between the lines.

3. Jul 30, 2011

### HallsofIvy

Staff Emeritus
Let the two lines be l1 and l2. Choose any point, P, on l1 and construct the plane perpendicular to l1 through P. Find the point, Q, where l2 intersects that plane. Find the distance between P and Q.