How do I find the distance between two lines passing through different points?

[SteliosVas]

Homework Statement

Okay thanks everyone for taking the time to read this quesiton. Now I sort of know how to do this question but I am not entirely sure how I should approach it:

I have points A(2,1,2) B(3,2,-1) C(-1,2,4) and D(-2,1,3)

Now a line passes through points A and B and also C and D respectively. What essentially we need to find is that the line that passes through A and B is at least 1 metre apart from the line that passes through C and D

Homework Equations

Dot Product, cross product. unit vector, shortest distance between two points*

The Attempt at a Solution

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So in order for there to be a distance we must check if the lines are parallel or not...
After conducting a test I found that these lines are skewed.

What I am confused about is this. I have 4 sets of coordinates now I know if I do

AB = B - A = (1,1,-1)

and CD = D - C = (-1,-1,-1) I get two vector equations

Now to find the distance between two vector equations we use vector projection and the unit vector but I am not sure how I can lay that out

 

[BvU]

So you have two lines $\vec A + \lambda(\vec B - \vec A)$ and $\vec C + \mu (\vec D - \vec C)$.
If I follow this example I need the veector product $\vec N = \vec {AB} \times \vec {CD}$ as the main step.

And note their extremely important tip: make a picture to see what you are doing !