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Let V → θ = cos ⁡ θ + sec ⁡ θ a ^ + cos ⁡ θ - s e c θ b ^ , where a ^ and b ^ are unit vectors and the angle between a ^ and b ^ is 60 o , then the minimum value of V → 4 is equal to

Correct answer

12

Step-by-step solution

V → 2 = cos ⁡ θ + sec ⁡ θ a ^ + cos ⁡ θ - sec ⁡ θ b ^ 2 = cos ⁡ θ + sec ⁡ θ 2 + cos ⁡ θ - sec ⁡ θ 2 + ( cos ⁡ θ - sec ⁡ θ ) ( cos ⁡ θ + sec ⁡ θ ) = 3 cos 2 ⁡ θ + sec 2 ⁡ θ ≥ 2 3 ⇒ least value of V → 2 = 2 3 ⇒ least value of V → 4 = 12

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