The shortest distance between the lines r =(1-t) +(t-2) +(3-2 t) k and r =(p+1) +(2 p-1) +(2 p+1) k is

Step-by-step solution

aligned ₁: r &=(1- t ) i +( t -2) j +(3-2 t ) k &=( i -2 j +3 k )+ t (- i + j -2 k ) ₂: r &=( i - j + k )+ p ( i +2 j +2 k ) Here & a ₂- a ₁=( i - j + k )-( i -2 j +3 k )= j -2 k & b ₁ b ₂= | array ccc i & j & k -1 & 1 & -2 1 & 2 & 2 array |= i (2+4)- j (0)-3 k =6 i -3 k aligned | b ₁ b ₂ | = 36+9 =3 5 shortest distance = | ( b ₁ b ₂ ) ( a ₂- a ₁ ) ( b ₁ b ₂ ) |= | (6 i -3 k ) ( j -2 k ) 3 5 |= 6 3 5 = 2 5