The shortest distance between the lines r = ( 1 3 i +2 j + 8 3 k )+ (2 i -5 j +6 k ) and r = (- 2 3 i - 1 3 k )+ ( j - k ) , , R , is:

Step-by-step solution

The given lines are r = a ₁ + b ₁ and r = a ₂ + b ₂ , where: a ₁ = 1 3 i + 2 j + 8 3 k b ₁ = 2 i - 5 j + 6 k a ₂ = - 2 3 i - 1 3 k b ₂ = j - k The shortest distance between two skew lines is given by d = |( a ₂ - a ₁) ( b ₁ b ₂)| | b ₁ b ₂| . First, finding the vector a ₂ - a ₁ : a ₂ - a ₁ = (- 2 3 - 1 3 ) i + (0 - 2) j + (- 1 3 - 8 3 ) k = - i - 2 j - 3 k Next, finding the cross product b ₁ b ₂ : b ₁ b ₂ = vmatrix i & j & k 2 & -5 & 6 0 & 1 & -1 vmatrix = i (5 - 6) - j (-2 - 0) + k (2 - 0) = - i + 2 j + 2 k The ma