If $\alpha, \beta$ and $\gamma$ are non-zero vectors such that $|\beta|=|\gamma|=1$ and $|\alpha|=10$, then…

If $\alpha, \beta$ and $\gamma$ are non-zero vectors such that $|\beta|=|\gamma|=1$ and $|\alpha|=10$, then $(\alpha \times(\beta+\gamma) \times(\beta \times \gamma) \cdot(\beta-\gamma)=$
  1. 10
  2. 1
  3. 0
  4. 12

Solution

$\begin{aligned} & (\alpha \times(\beta+\gamma)) \times(\beta \times \gamma) \cdot(\beta-\gamma) \\ = & (\alpha \times(\beta+\gamma)) \cdot[(\beta \times \gamma) \times(\beta-\gamma)] \\ = & (\alpha \times(\beta+\gamma)) \cdot[(\beta \cdot(\beta-\gamma) \gamma-(\gamma \cdot(\beta-\gamma) \beta] \\ = & (\alpha \times(\beta+\gamma)) \cdot\left[\left(\left.\beta\right|^2-\beta \cdot \gamma\right) \gamma-\left(\gamma \cdot \beta-|\gamma|^2\right) \beta\right] \\ = & (\alpha \times(\beta+\gamma)) \cdot[(1-\beta \cdot \gamma)](\gamma+\beta] \\ = & (1-\beta \cdot \gamma)[\alpha \times \beta) \cdot \gamma+(\alpha \times \gamma) \cdot \beta] \\ \Rightarrow & (1-\beta \cdot \gamma)[(\alpha \times \beta) \cdot \gamma-(\alpha \times \beta) \cdot \gamma]=0\end{aligned}$

Asked in: AP EAMCET 2018 (22 Apr Shift 2)

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