Find the shortest distance between the lines (x - 3)/3 = (y - 8)/-1

1.	Find the shortest distance between the lines (x - 3)/3 = (y - 8)/-1 = (z - 3)/1 and (x + 3)/-3 = (y + 7)/2 = (z - 6)/4.
Answer» Given the line is (x - 3)/3 = (y - 8)/-1 = (z - 3)/1 (x + 3)/-3 = (y + 7)/2 = (z - 6)/4 The standard equation of a line x₁ = 3, y₁ = 8, z₁ = 3 x₂= -3, y₂ = -7, z₂ = 6 Thus, the shortest distance d = \|(x₂ - x₁,y₂ - y₁,z₂ - z₁),(a₁,b₁,c₁),(a₂,b₂,c₂)\| = \|(6,15,-3),(3,-1,1),(-3,2,4)\|/√((-6)² + (15)² + (3)²) = (6(-4 - 2) -15(12 + 3) - 3(6 - 3))/√(36 + 225 + 9) = (-36 - 225 - 9)/√270 = -270/√270 = -√270

Find the shortest distance between the lines (x - 3)/3 = (y - 8)/-1 = (z - 3)/1 and (x + 3)/-3 = (y + 7)/2 = (z - 6)/4.

Answer»

Given the line is

(x - 3)/3 = (y - 8)/-1 = (z - 3)/1

(x + 3)/-3 = (y + 7)/2 = (z - 6)/4

The standard equation of a line

x₁ = 3, y₁ = 8, z₁ = 3

x₂= -3, y₂ = -7, z₂ = 6

Thus, the shortest distance

d = |(x₂ - x₁,y₂ - y₁,z₂ - z₁),(a₁,b₁,c₁),(a₂,b₂,c₂)|

= |(6,15,-3),(3,-1,1),(-3,2,4)|/√((-6)² + (15)² + (3)²)

= (6(-4 - 2) -15(12 + 3) - 3(6 - 3))/√(36 + 225 + 9)

= (-36 - 225 - 9)/√270 = -270/√270 = -√270