Finding the distance between two parametric lines

[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.

 

[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.

 

[ehild]

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).

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