Finding Equation of an Orthogonal Line

[TranscendArcu]

Homework Statement

Let L₁ be the line (0,4,5) + <1,2,-1>t and L₂ be the line (-10,9,17) + <-11,3,1>t

a) Find the line L passing through and orthogonal to L₁ and L₂
b) What is the distance between L₁ and L₂

The Attempt at a Solution

I only know how to do part of part a). I can only find the direction vector of the orthogonal line by taking the cross product. I have,

<1,2,-1> x <-11,3,1> = <5,10,25>, which I simplify to <1,2,5>.

It doesn't seem very obvious to me how I can find a point (presumably on either L₁ or L₂) such that a line containing this point, pointing in the direction of <1,2,5>, passes through both L₁ and L₂.

For part b), I presume that these two lines are skew. (How do you check if lines are parallel or intersect?) The distance between these two lines is

|<5,10,25> dot [(0,4,5) - (-10,9,17)]| = |<5,10,25> dot <10,-5,-12>| = |50 - 50 - 300| = 300. So the distance is 300?

 

[TranscendArcu]

Actually, I think I did the distance calculation wrong. What I actually have is 60/sqrt(30)

 

[SammyS]

Actually, I think I did the distance calculation wrong. What I actually have is 60/sqrt(30)

Looks good.

$\displaystyle \frac{60}{\sqrt{30}}=2\sqrt{30}\,$

 

[TranscendArcu]

Okay, well it's good to know I can do part b. But what I think I really don't understand is how to do part a. One of my friends said it could be done if one has already calculated the distance, but this seems like doing the problem backwards, which I would like to avoid if possible.