Question

Let \(\hat{\alpha}, \hat{\beta}, \hat{\gamma}\) be three unit vectors such that \(\hat{\alpha} \times(\hat{\beta} \times \hat{\gamma})=\frac{1}{2}(\hat{\beta}+\hat{\gamma})\) where \(\hat{\alpha} \times(\hat{\beta} \times \hat{\gamma})=(\hat{\alpha} \cdot \hat{\gamma}) \hat{\beta}-(\hat{\alpha} . \hat{\beta}) \hat{\gamma}\). If \(\hat{\beta}\) is not paralle to \(\hat{\gamma}\), then the angle between \(\hat{\alpha}\) and \(\hat{\beta}\) is
(A) \(\frac{5 \pi}{6}\)
(B) \(\frac{\pi}{6}\)
(C) \(\frac{\pi}{3}\)
(D) \(\frac{2 \pi}{3}\)

Answer 1

Ans: (D)
Hint : \(\vec{\alpha} \times(\vec{\beta} \times \vec{\gamma})=\frac{1}{2}(\vec{\beta}+\vec{\gamma}),(\vec{\alpha} \cdot \vec{\gamma}) \vec{\beta}-(\vec{\alpha} \cdot \vec{\beta}) \vec{\gamma}-=\frac{1}{2}(\vec{\beta}+\vec{\gamma}),-(\vec{\alpha} \cdot \vec{\beta})=\frac{1}{2},-\cos \theta=\frac{1}{2}, \cos \theta=\frac{-1}{2}, \theta=120^{\circ}=\frac{2 \pi}{3}\)

Answer 2

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Answer 3

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