If a → = 2 i ^ + j ^ + 3 k ^ , b → = 3 i ^ + 3 j ^ + k ^ and c → = c 1 i ^ + c 2 j ^ + c 3 k ^ are coplanar vectors and a → · c → = 5 , b → ⊥ c → , then 122 c 1 + c 2 + c 3 is equal to ______.

Step-by-step solution

As the given vectors are coplanar, so 2 1 3 3 3 1 c 1 c 2 c 3 = 0 ⇒ 8 c 1 - 7 c 2 - 3 c 3 = 0       . . . i Also, a → · c → = 5 ⇒ 2 c 1 + c 2 + 3 c 3 = 5       . . . i i and b → · c → = 0 ⇒ 3 c 1 + 3 c 2 + c 3 = 0       . . . i i i Solving the above three equations using Cramer's rule, we get, ∆ = 8 - 7 - 3 2 1 3 3 3 1 = - 122 ∆ 1 = 0 - 7 - 3 5 1 3 0 3 1 = - 10 ∆ 2 = 8 0 - 3 2 5 3 3 0 1 = 85 &#8710