Shortest distance between the lines r =(8+3 ) -(9+16 ) +(10+7 ) k and r =15 +29 +5 k + (3 +8 -5 k ) is

Step-by-step solution

The given lines are r = a₁ + b₁ and r = a₂ + b₂ , where: a₁ = 8 i - 9 j + 10 k , b₁ = 3 i - 16 j + 7 k a₂ = 15 i + 29 j + 5 k , b₂ = 3 i + 8 j - 5 k The shortest distance d between two skew lines is given by d = | ( a₂ - a₁ ) ( b₁ b₂ ) | b₁ b₂ | | . First, calculate a₂ - a₁ = (15-8) i + (29 - (-9)) j + (5-10) k = 7 i + 38 j - 5 k . Next, calculate the cross product b₁ b₂ : b₁ b₂ = vmatrix i & j & k 3 & -16 & 7 3 & 8 & -5 vmatrix = i (80 - 56) - j (-15 - 21) + k (24 - (-48)) = 24 i + 36 j + 72 k = 12(2 i + 3 j + 6 k