# Finding the distance between two parametric lines

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1. Oct 21, 2014

### mnmakrets

1. Write down the equation for the line in 3D through the point a=(1,2,4), parallel to the line r=(1,-5,0)+λ(1,2,2). Then, find the distance between these lines.

2.

3. Lets say, b= (1,2,2). b is parallel to given line, so it must also be parallel to the new line.
My guess is that the equation of the new line is then;
c=(1,2,4)+λ(1,2,2).

I don't know how to approach the rest of the problem, this is a new topic for me, however this is revision for many students in my class so my teacher did not explain this thoroughly, i would greatly appreciate any hints for this problem, and/or any useful webpages that would help me here. Thanks in advance.

2. Oct 21, 2014

### RUber

Assuming you mean Euclidean distance, the minimum distance between 2 parallel lines is the length of a perpendicular segment connecting them.
The cross product of 2 vectors will be perpendicular to them both.

3. Oct 21, 2014

### ehild

It is correct so far, but use some other letter inside of lambda in the equation of the new line.
When trying to solve such problems, it is very useful to make a figure. See the one attached.
You need a line which intersects both parallel lines and perpendicular to them.
The lines have common normal planes. Their points of intersection with such a plane are D distance apart, and D is the distance between the lines.

ehild