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Angle between Two Line Segments

If a⃗, b⃗ a...

Question

If $\to a, \to b$ are two unit perpendicular vectors and $\to c$ is a unit vector which is inclined an angle $θ$ with $\to a$ and $\to b$ then $\to c = α ¯ ¯ ¯ a + β ¯ ¯ b + γ (\to a \times \to b)$ , then $γ =$

A

cos 2 θ

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B

- cos 2 θ

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C

sin 2 θ

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D

- sin 2 θ

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Solution

The correct option is B $- cos 2 θ$
${\to c}^{2} = α \to a . \to c + β \to b . \to c + γ \to c . (\to a \times \to b)$
$1 = cos θ α + cos θ β + x [\to a + \to b + \to c]$ --------(1)
$\to a . \to c = α$
$cos θ = α$ -------------(2)
$\to b . \to c = β$
$cos θ = β$ ---------------(3)
${sin}^{2} θ - {cos}^{2} θ = γ \to a \to b \to c$
$- cos θ = γ$

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