I'm a bit uncertain about this question and would like some help, as I don't have the correct answer. Have I done this correctly?(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

What is the shortest distance between the two lines A = (1,2,3) + t(0,1,1) and B = (1,1,1) + s(2,3,1)

3. The attempt at a solution

My reasoning: The vector AB is shortest when AB is orthogonal to BOTH A and B.

Therefore the scalar product AB [tex]\circ[/tex] A = AB [tex]\circ[/tex] B = 0. That gives a system with two equations

AB = (2s, -1+3s-t, -2+s-t)

AB [tex]\circ[/tex] A = -3+4s-2t=0

AB [tex]\circ[/tex] B = 14s-5-4t=0

which when solved gives s = -1/6 and t = -11/6.

I now seek |AB|, or the LENGTH of the vector.

Substituting s and t with the corresponding values and then using Pythagoras gives:

|AB| = sqrt(1/3)

Is this correct? Is there perhaps an easier way to do this?

Danke schÃ¶n!

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# Shortest distance between two lines

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