Let θ be the angle between two unit vectors \(\rm \vec a\ and\ \vec b. \ if\ \vec a+2\vec b\) is perpendicular to \(\rm 5\vec a-4\vec b\) then what is cos θ + cos 2θ equal to?

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This question was previously asked in

NDA-II 2024 (Maths) Official Paper (Held On: 01 Sept, 2024)

Answer 1

Explanation:

Given:

\(⃗ a+2⃗ b\) and \(\rm 5⃗ a-4⃗ b\) are perpendicular vectors.

⇒ \((⃗ a+2⃗ b).\)\((\rm 5⃗ a-4⃗ b)\) = 0

⇒ \(5|⃗ a|^2 -4⃗ a.⃗ b + 10⃗ a . ⃗ b - 8(⃗ b)^2=0\)

⇒ \(5× 1 + 6⃗ a.⃗ b-8 =0\)

Now \(\vec a, \vec b\) are unit vectors

⇒ \(6 |\vec a||\vec b| Cosθ = 3\)

⇒ 1.1 cosθ =1/2

⇒Cosθ = 1/2

Now

cos2 θ = Cos²θ -1

= 2× 1/4 -1

= \(-\frac{1}{2}\)

Now,

cosθ + cos2θ = 1/2 -1/2 =0

∴The Correct answer is Option a

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