Find pair of 3D points given distance.

Question

In a problem, I was given two lines AB going through A(5, -1, 2) and B(2, -1, 4) and CD going through C(1, 2, 6) and D(0, 1, 2). The distance between these lines is 38/sqrt(209). I am supposed to find a unique pair of points that lie on lines AB and CD and the distance between these points is also 38/sqrt(209).

Answer 1

The problem is known as finding the shortest distance between two skew lines and the points on the two lines

There are several approaches to solve this problem.

It can be done with knowledge of Analytic Geometry up to methods using multivariable calculus.

APPROACHE 1 USING MULTIVARIATE CALCULUS.

1. Easily can be shown that the two lines are skew
2. The directing vectors of the line passing through A and B and line passing through D and C are respectfully u =< −3, 0, 2 > and v = <1, 1, 4 >
3. The parametric equation for line one are x = 5 −3t, y=−1 , z=2 +2t Equation (1)
4. The parametric equation for line two are x = k, y=1 +k , z=2 +4k Equation (2)
5. The distance between to two points on the above lines is D=√[ (k+3t−5)² +(k+2)² +(4k-2t)²]

To minimize D suffices to minimize the quantity Ω(t,k)= (k+3t−5)² +(k+2)² +(4k-2t)² ⇒

Ω(t,k)= 18k²+13t²-10kt-6k-30t+29

Then we need the second derivative test for functions of two variables of the function Ω.

Ω_t= 26t-2k-30 and Ω_k=36k-2t-6

Solving the system 26t-2k-30=0 and 36k-2t-6=0 we find t=285/ 209 and k=84/209.

Substituting these values back to Equation (1) and Equation (2) we get the points

(x₁, y₁,z₁)= (10/11, -1, 52/11) and (x₂, y₂,z₂) =(84/209, 293/209, 754/209)

APPROACH 2 USING ANALYTIC GEOMETRY

1. Lines L₁ and L₂ belong to two parallel planes P₁ and P₂. Find the equations of those two planes .The normal vector for both planes is the cross product of the directing vectors of the two lines.
2. Find the equation of the plane P₃ containing L₁ perpendicular to P₂.
3. Find the equation of the plane P₄ containing L₂ perpendicular to P₁.
4. Find parametric equations of line L₃ which is the intersection of planes P₃ and P₄
5. The points of interest are the intersections of L₃ with lines L₁ and L₂