If the volume of a parallelopiped, whose coterminous edges are given by the vectors a → = i ^ + j ^ + n k ^ , b → = 2 i ^ + 4 j ^ - n k ^ and, c → = i ^ + n j ^ + 3 k ^ ( n ≥ 0 ) is 158 cubic units, then :

Step-by-step solution

The volume of the parallelepiped is, v = a →   b →   c → ⇒ 1 1 n 2 4 - n 1 n 3 = ± 158 ⇒ 1 12 + n 2 - 1 ( 6 + n ) + n ( 2 n - 4 ) = ± 158 ⇒ 3 n 2 - 5 n - 152 = 0 or 3 n 2 - 5 n + 164 = 0 ( ∵   D < 0 , no real roots of 3 n 2 - 5 n + 164 = 0 ) n = 8 ,   - 19 3    ⇒    n = 8 , as n ≥ 0 then,   b → · c → = 2 + 4 n - 3 n = 10 a → · c → = 1 + n + 3 n = 33