Let , , be three unit vectors such that ( )=1 2 ( ) where ( )=( )… - Vidyadip

MCQ

Let $\hat{\alpha}, \hat{\beta}, \hat{\gamma}$ be three unit vectors such that $\hat{\alpha} \times(\hat{\beta} \times \hat{\gamma})=\frac{1}{2}(\hat{\beta} \times \hat{\gamma})$ where $\hat{\alpha} \times(\hat{\beta} \times \hat{\gamma})=(\hat{\alpha} \cdot \hat{\gamma}) \hat{\beta}-(\hat{\alpha} \cdot \hat{\beta}) \hat{\gamma}$. If $\hat{\beta}$ is not parallel to $\hat{\gamma}$, then the angle between $\hat{\alpha}$ and $\hat{\beta}$ is

A
$\frac{5 \pi}{6}$
B
$\frac{\pi}{6}$
C
$\frac{\pi}{3}$
✓
$\frac{2 \pi}{3}$

✓

Answer

Correct option: D.

$\frac{2 \pi}{3}$

(D) $(\hat{\alpha} \cdot \hat{\gamma}) \hat{\beta}-(\hat{\alpha} \cdot \hat{\beta}) \hat{\gamma}=\frac{1}{2} \hat{\beta}+\frac{1}{2} \hat{\gamma}$
As $\hat{\beta}$ is not parallel to $\hat{\gamma}$,
$\hat{\alpha} \cdot \hat{\beta}=-\frac{1}{2}$
$\Rightarrow|\hat{\alpha}||\hat{\beta}| \cos \theta=-\frac{1}{2}$
$\Rightarrow \cos \theta=-\frac{1}{2}$ $\ldots[\because|\hat{\alpha}|=1,|\hat{\beta}|=1]$
$\Rightarrow \theta=\frac{2 \pi}{3}$

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