Question

Let $\vec{\alpha} . \vec{\beta} , \vec{\gamma}$ be three unit vectors such that $\vec{\alpha} . \vec{\beta} = \vec{\alpha} . \vec{\gamma} = 0$ and the angle between $\vec{\beta}$ and $\vec{\gamma}$ is $30^{\circ}$ . Then $\vec{\alpha}$ is

• A. $2 ( \vec{\beta} \times \vec{\gamma})$
• B. $- 2 ( \vec{\beta} \times \vec{\gamma})$
• C. $\pm 2 ( \vec{\beta} \times \vec{\gamma})$
• D. $( \vec{\beta} \times \vec{\gamma})$

Answer 1

Answer: (C)

$\pm 2 ( \vec{\beta} \times \vec{\gamma})$

Answer 2

$\vec{\alpha}=\lambda(\vec{\beta} \times \vec{\gamma})=\lambda\left(\vec{\beta}|| \vec{\gamma}|| \sin 30^{\circ} \mid\right.$
$\Rightarrow |\vec{\alpha}|=|\lambda|\left(|\beta||\vec{\gamma}| \cdot \frac{1}{2}\right)$
$\Rightarrow 1=|\lambda| \cdot 1 \cdot 1 \cdot \frac{1}{2}$
$\Rightarrow |\lambda|=2$
$\Rightarrow \lambda=\pm 2$
$\therefore \vec{\alpha}=\pm 2(\vec{\beta} \times \vec{\gamma})$