If p is the shortest distance between the lines x+1 7 = y+1 -6 = z+1 and r = (3 i + 5 j + 7 k ) + ( i - 2 j + k ) then [p] is... ,(where [ ,. ,] denotes the greatest integer function.)

Step-by-step solution

The shortest distance p between two lines r = a ₁ + b ₁ and r = a ₂ + b ₂ is given by p = |( a ₂ - a ₁) ( b ₁ b ₂)| | b ₁ b ₂| . For the first line, a ₁ = - i - j - k and b ₁ = 7 i - 6 j + k . For the second line, a ₂ = 3 i + 5 j + 7 k and b ₂ = i - 2 j + k . a ₂ - a ₁ = 4 i + 6 j + 8 k b ₁ b ₂ = vmatrix i & j & k 7 & -6 & 1 1 & -2 & 1 vmatrix = -4 i - 6 j - 8 k ( a ₂ - a ₁) ( b ₁ b ₂) = (4)(-4) + (6)(-6) + (8)(-8) = -16 - 36 - 64 = -116 | b ₁ b ₂| = (-4)^2 + (-6)^2 + (-8)^2 = 16 + 36 + 64 = 116 p = |-116| 116 = 11