Consider three vectors V → 1 = sin ⁡ θ i ^ + cos ⁡ θ j ^ + a - 3 k ^ , V → 2 = sin ⁡ θ + cos ⁡ θ i ^ + cos ⁡ θ - sin ⁡ θ j ^ + b - 4 k ^ and V → 3 = cos ⁡ θ i ^ + sin ⁡ θ j ^ + c - 5 k ^ . If the resultant of V → 1 ,   V → 2 and V → 3 is equal to λ i ^ , where θ ∈ - π ,

Step-by-step solution

Resultant of V → 1 , V → 2 , V → 3 is V → 1 + V → 2 + V → 3 = λ i ^ ⇒ 2 sin ⁡ θ + 2 cos ⁡ θ i ^ + 2 cos ⁡ θ - 2 sin ⁡ θ j ^ + a + b + c - 12 k ^ = λ i ^ On comparison, we get, 2 sin ⁡ θ + 2 cos ⁡ θ = λ 2 cos ⁡ θ - 2 sin ⁡ θ = 0 ⇒ tan ⁡ θ = 1 ⇒ number of values of θ in - π , π is 2 Also, a + b + c = 12 ⇒ number of solutions = 11 C