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?