If the volume of the parallelopiped formed by the vectors i ^ + a j ^ + k ^ , j ^ + a k ^ and a i ^ + k ^ becomes minimum, then a =

Step-by-step solution

The Volume of parallelopiped is given by a →   b →   c → , where a → ,   b → ,   c → represents the vectors that form the parallelopiped. Putting the given vectors, we get ⇒ V = 1 a 1 0 1 a a 0 1 = 1 - a - a 2 - a = 1 + a 3 - a For minimum value d V d a = 0 ⇒ d V d a = 3 a 2 - 1 = 0 ⇒ a = ± 1 3 Now, at a = 1 3 ⇒ d 2 V d a 2 = 6 a ⇒ d 2 V d a 2 = 6 3 > 0 Therefore, Volume V is minimum at a = 1 3 .