If d 1 is the shortest distance between the lines x + 1 = 2 y = - 12 z , x = y + 2 = 6 z - 6 and d 2 is the shortest distance between the lines x - 1 2 = y + 8 - 7 = z - 4 5 , x - 1 2 = y - 2 1 = z - 6 - 3 , then the value of 32 3 d 1 d 2 is :

Step-by-step solution

Let, L 1 : x + 1 1 = y 1 2 = z - 1 12 , And L 2 : x 1 = y + 2 1 = z - 1 1 6 Now, given d 1 = shortest distance between L 1 and L 2 Now using the formula of shortest distance we get, d 1 = a → 2 - a → 1 · b → 1 × b → 2 b → 1 × b → 2 ⇒ d 1 = i ^ - 2 j ^ + k ^ · i ^ + 1 2 j ^ - 1 12 k ^ × i ^ + j ^ + 1 6 k ^ i ^ + 1 2 j ^ - 1 12 k ^ × i ^ + j ^ + 1 6 k ^ ⇒ d 1 = i ^ - 2 j ^ + k ^ · i ^ 6 - j ^ 4 + k ^ 2 i ^ 6 - j ^ 4 + k ^ 2 ⇒ d 1 = 1 6 + 1 2 + 1 2 49 144 ⇒ d 1 = 7 6 7 12 = 2 Now, let L 3 : x - 1 2 = y + 8 - 7 = z - 4