Question

Let L₁ be the line passing through the point P₁(?5, ?3, ?2) with direction vector d=[0, ?3, ?2]^T, and let L₂ be the line passing through the point P₂(?2, 3, ?3) with the same direction vector.
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 '?' where needed to give an exact value for your answer.

d =

0

Q₁ = (

0

,

0

,

0

)Q₂ = (

0

,

0

,

0

)

Answer 1

Given L₁ is the line passing through the point P₁(-5, -3, -2) with direction vector d=[0, -3, -2]^T, and let L₂ be the line passing through the point P₂(-2, 3, -3) with the same direction vector.

Then the equation of the line L₁ is :

And, the equation of the line L₂ is :

Then, the shortest distance is, d = = = 0.

Given that Q₁ and Q₂ are points on the line L₁ and L₂ respectively.

Then, the coordinates of Q₁ and Q₂ are (-5,-3-3r,-2-2r) and (-2,3-3s,-3-2s).

Now, the direction ratios of the line Q₁Q₂ are (-2+5,3-3s+3+3r,-3-2s+2+2r) ,i.e., (3,6-3s+3r,-1-2s+2r).

Since the line Q₁Q₂ is perpendicular to the lines L₁ and L₂, then we have,

0*(3)+(-3)*(6-3s+3r)+(-2)*(-1-2s+2r) = 0

i.e., -18+9s-9r+2+4s-4r = 0

i.e., -16+13s-13r = 0

i.e., s = r+(16/13)