Find the shortest distance between the following pair of lines: x + 1 7 = y + 1 - 6 = z + 1 1 x - 3 1 = y - 5 - 2 = z - 7 1 .

Given: ⇒ x + 1 7 = y + 1 - 6 = z + 1 1 ⇒ x - 3 1 = y - 5 - 2 = z - 7 1 Comparing the above equations with the standard form, we get, ⇒ x - x 1 a 1 = y - y 1 b 1 = z - z 1 c 1 ⇒ x - x 2 a 2 = y - y 2 b 2 = z - z 2 b 2 We get, x 1 = - 1 , y 1 = - 1 , z 1 = - 1 , a 1 = 7 , b 1 = - 6 , c 1 = 1 And, x 2 = 3 , y 2 = 5 , z 2 = 7 , a 2 = 1 , b 2 = - 2 , c 2 = 1 The shortest distance between the two lines can be given as, x - x 1 a 1 = y - y 1 b 1 = z - z 1 c 1 x - x 2 a 2 = y - y 2 b 2 = z - z 2 b 2 ⇒ d = x 2 - x 1 y 2 - y 1 z 2 - z 1 a 1 b 1 c 1 a 2 b 2 c 2 b 1 c 2 - b 2 c 1 2 + c 1 a 2 - c 2 a 1 2 + a 1 b 2 - a 2 b 1 2 Then, we get, ⇒ x 2 - x 1 y 2 - y 1 z 2 - z 1 a 1 b 1 c 1 a 2 b 2 c 2 = 3 - - 1 5 - - 1 7 - - 1 7 - 6 1 1 - 2 1 ⇒ 4 6 8 7 - 6 1 1 - 2 1 = 4 - 6 + 2 - 6 7 - 1 + 8 - 14 + 6 ⇒ - 16 - 36 - 64 = - 116 And, ⇒ b 1 c 2 - b 2 c 1 2 + c 1 a 2 - c 2 a 1 2 + a 1 b 2 - a 2 b 1 2 = - 6 + 2 2 + 1 - 7 2 + - 14 + 6 2 = 16 + 36 + 64 = 116 = 2 29 Now, we get, ⇒ d = - 116 2 29 d = 58 29 d = 2 29

Find the shortest distance between the lines x - 1 2 = y - 2 3 = z - 3 4 and x - 2 3 = y - 4 4 = z - 5 5 .

Given: ⇒ x - 1 2 = y - 2 3 = z - 3 4 x - 2 3 = y - 4 4 = z - 5 5 Comparing the equations with the standard cartesian form, we get, ⇒ x - x 1 a 1 = y - y 1 b 1 = z - z 1 c 1 x - x 2 a 2 = y - y 2 b 2 = z - z 2 c 2 We get, x 1 = 1 , y 1 = 2 , z 1 = 3 , a 1 = 2 , b 1 = 3 , c 1 = 4 And, x 2 = 2 , y 2 = 4 , z 2 = 5 , a 2 = 3 , b 2 = 4 , c 2 = 5 The shortest distance between the lines can be given as, ⇒ d = x 2 - x 1 y 2 - y 1 z 2 - z 1 a 1 b 1 c 1 a 2 b 2 c 2 b 1 c 2 - b 2 c 1 2 + c 1 a 2 - c 2 a 1 2 + a 1 b 2 - a 2 b 1 2 Now, we get, x 2 - x 1 y 2 - y 1 z 2 - z 1 a 1 b 1 c 1 a 2 b 2 c 2 = 2 - 1 4 - 2 5 - 3 2 3 4 3 4 5 ⇒ 1 2 2 2 3 4 3 4 5 = 1 15 - 16 - 2 10 - 12 + 2 8 - 9 ⇒ - 1 + 4 - 2 = 1 Now, we get, b 1 c 2 - b 2 c 1 2 + c 1 a 2 - c 2 a 1 2 + a 1 b 2 - a 2 b 1 2 = ( 15 - 16 ) 2 + ( 12 - 10 ) 2 + ( 8 - 9 ) 2 = - 1 2 + 2 2 + - 1 2 = 6 The shortest distance between the lines can be given as, ⇒ d = 1 6 d = 1 6