Find the length and the equations of the line of shortest distance between the lines given by:

and

Given : Cartesian equations of lines

Formulae :

1. Condition for perpendicularity :

If line L1 has direction ratios (a₁, a₂, a₃) and that of line L2 are (b₁, b₂, b₃) then lines L1 and L2 will be perpendicular to each other if

2. Distance formula :

Distance between two points A≡(a₁, a₂, a₃) and B≡(b₁, b₂, b₃) is given by,

3. Equation of line :

Equation of line passing through points A≡(x₁, y₁, z₁) and B≡(x₂, y₂, z₂) is given by,

Answer :

Given equations of lines

Direction ratios of L1 and L2 are (3, -1, 1) and (-3, 2, 4) respectively.

Let, general point on line L1 is P≡(x₁, y₁, z₁)

x₁ = 3s+3 , y₁ = -s+8 , z₁ = s+3

and let, general point on line L2 is Q≡(x₂, y₂, z₂)

x₂ = -3t – 3 , y₂ = 2t – 7 , z₂ = 4t + 6

Direction ratios of are ((-3t – 3s - 6), (2t + s - 15), (4t – s + 3))

PQ will be the shortest distance if it perpendicular to both the given lines

Therefore, by the condition of perpendicularity,

3(-3t – 3s - 6) – 1(2t + s - 15) + 1(4t – s + 3) = 0 and

-3(-3t – 3s - 6) + 2(2t + s - 15) + 4(4t – s + 3) = 0

⇒ -9t – 9s – 18 – 2t – s + 15 + 4t – s + 3 = 0 and

9t + 9s + 18 + 4t + 2s – 30 + 16t – 4s + 12 = 0

⇒ -7t – 11s = 0 and

29t + 7s = 0

Solving above two equations, we get,

t = 0 and s = 0

therefore,

P ≡ (3, 8, 3) and Q ≡ (-3, -7, 6)

Now, distance between points P and Q is

Therefore, the shortest distance between two given lines is

Now, equation of line passing through points P and Q is,

Therefore, equation of line of shortest distance between two given lines is