Let L1 be the line passing through the point P1 4 2 1 with

Let L_1 be the line passing through the point P_1 = (4, 2, 1) with direction vector 3=[2, -3, 2]^T, and let L_2 be the line passing through the point P_2 = (-3, -5, 5) with the same direction vector. Find the shortest distance d between these two lines, and find a point Q_1 on L_1 and a point Q_2 on L_2 so that d(Q_1, Q_2) = d. Use the squareroot symbol squareroot where needed to give an exact value for your answer. d= 0 Q_1 = (0, 0, 0) Q_2 = (0, 0, 0)

Solution

We know that the equation of a line through the point A (x₁, y₁, z₁) and the direction vector (l, m, n) is

(x -x₁)/l = (y-y₁)/m = (z-z₁)/n = k (say)

Then a on this line at a distance ‘k’ from the point A (x₁, y₁, z₁) is P (x₁ + l k, y₁ + m k, z₁ + n k).

In view of the above, the cartesian equations of the lines L₁ and L₂ are (x-4)/2 = (y-2)/-3 = (z-1)/2 = r₁ (say) …(1)

and (x +3)/2=( y+5)/-3 = (z-5)/2 = r₂ (say)…(2).

Then a point Q₁ on the line L₁ is (4+2r₁,2-3r₂,1+2r₁) and a point Q₂ on the line L₂ is (-3+2r₂,-5-3r₂,5+2r₂)   

If Q₁Q₂ is the line of shortest distance, Its direction ratios will be [(4+2r₁+3-2r₂),(2-3r₁+5+3r₂), (1+2r₁ – 5-2r₂ )]

or, (7+2r₁ -2r₂ , 7-3r₁ +3r₂ , -4+2r₁ -2r₂ ).This line Q₁Q₂ is perpendicular to both the lines L₁and L₂ so that (2,-3,2).( 7+2r₁ -2r₂ , 7-3r₁ +3r₂ , -4+2r₁ -2r₂) = 0, or, 2(7+2r₁ -2r₂) -3(7-3r₁ +3r₂) +2(-4+2r₁ -2r₂) = 0 or, 14+ 4r₁ -4r₂ -21+9r₁ -9r₂ -8 +4r₁ -4r₂ = 0 or, -15 + 17r₁ – 17r₂ =0 or, 17(r₁ –r₂ ) = 15 or r₂ = r₁ - 15/17.Then the direction ratios of the line Q₁Q₂ are ( 7+2r₁ -2r₁ +30/17, 7-3r₁ +3r₁ -45/17, -4+2r₁ -2r₁+30/17) or, (149/17, 74/17, -38/17) The norm of Q₁ Q₂ is [ (149/17)² + (74/17)² + (-38/17)² ] = (1/17) ( 22201+5476 + 1444) = (1/17)29121 . Thus the shortest distance between the lines L and L is (29121)/17. The point Q₁ on the line L₁ is (4+2r₁,2-3r₂,1+2r₁) and the point Q₂ on the line L₂ is (-3+2r₂,-5-3r₂,5+2r₂) = (-3+2r₁ -30/17, -5-3r₁ +45/17, 5+2r₁ -30/17) =( -81/17 +2r₁ , -40/17-3r₁ , 55/17+2r₁ ) where r₁ is an arbitrary real number.