The shortest distance between the lines, where the first line passes through (0,0,0) and (2,0,3) and the second line passes through (2,5,0) and (0,4,0) is

Step-by-step solution

The equation of the first line passing through (0,0,0) and (2,0,3) is r = a₁ + b₁ , where a₁ = 0 i + 0 j + 0 k and b₁ = 2 i + 0 j + 3 k . The equation of the second line passing through (2,5,0) and (0,4,0) is r = a₂ + b₂ , where a₂ = 0 i + 4 j + 0 k and b₂ = -2 i - j + 0 k . The shortest distance d between two lines is given by d = |( a₂ - a₁ ) ( b₁ b₂ )| | b₁ b₂ | . We have a₂ - a₁ = 4 j . Now, b₁ b₂ = vmatrix i & j & k 2 & 0 & 3 -2 & -1 & 0 vmatrix = 3 i - 6 j - 2 k . The magnitude is | b₁ b₂ | = 3^2 + (-6)^2 + (