MHB [Linear Algebra] - Find the shortest distance d between two lines

[CoolMan2017]

Let L1 be the line passing through the point P1=(−2,−11,9) with direction vector d2=[0,2,−2]T, and let L2 be the line passing through the point P2=(−2,−1,11) with direction vector d2=[−1,0,−1]T Find the shortest distance d between these two lines, and find a point Q1 on L1 and a point Q2 on L2 so that d(Q1,Q2)=d. Use the square root symbol to get the exact value

 

[MarkFL]

Hello and welcome to MHB! :D

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[HallsofIvy]

Looks pretty straight forward to me- though, frankly, I don't see what it has to do with "Linear Algebra"- just basic Calculus.

Line L1 can be written in a vector equation as (x, y, z)= (-2, -11, 9)+ (0, 2, -2)t= (-2, -11+ 2t, 9- 2t) and L2 as (x, y, z)= (-2, -1, 11)+ (-1, 0, -1)s= (-2- s, -1, 11- s). The distance between any two points, one on the first line, the other on the second is $D= \sqrt{(-2- (-2-s))^2+ (-11+ 2t- (-1))^2+ (9- 2t- (11- s)^2}= \sqrt{s^2+ (-10+ 2t)^2+ (s- 2t- 2)^2}$.

Minimizing that distance is the same as minimizing its square:
$D^2= s^2+ (-10+ 2t)^2+ (s- 2t- 2)^2$.

Set the derivatives with respect to s and t to 0 and solve the two equations for s and t.