The shortest distance between the two lines L1: r = (1,0,0) + t(2,3,4) and L2: u = (2,1,0) + s(1,2,0) can be determined by constructing the function "distance squared between the lines," defined as the norm squared of the distance vector function d(s,t) = r(t) - u(s). This approach simplifies the problem, as the minimum distance occurs when the distance squared is minimized. The function is dependent on the variables t and s, and the extrema can be found using standard optimization techniques without needing the second derivative test.
PREREQUISITESMathematicians, physics students, engineers, and anyone involved in computational geometry or optimization problems will benefit from this discussion.
.quasar987 said:There is probably a shorter way in terms of vectors and stuff, but here's a way:
Build the function "distance squared btw the lines", defined as the norm squared of the distance vector function
[tex]\vec{d}(s,t) = \vec{r}(t) - \vec{u}(s)[/tex]
The reason we work with the function distance squared instead of the function distance itself is that the distance is minimum when the distance squared is minimum, and as you'll see, it is much easier to find when d² is minimum then when d is minimum.
This function is a function of the two variables t and s. Use the usual method to find the extrema of the function. (no need to use the second derivative test to determine the nature of each extrema; yuo know the function distance quared as no maxima and only one minimum