$\mathbf{a} \times [\mathbf{a} \times (\mathbf{a} \times \mathbf{b})]$ is equal to

$\mathbf{a} \times [\mathbf{a} \times (\mathbf{a} \times \mathbf{b})]$ is equal to
  1. $|\vec{a} \times \vec{a}| \cdot |\vec{b} \times \vec{a}|$
  2. $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{a}) - \mathbf{b} \cdot (\mathbf{a} \times \mathbf{b})$
  3. $\left[ \mathbf{a} \cdot (\mathbf{a} \times \mathbf{b}) \right] \mathbf{a}$
  4. $\left|\vec{a} \cdot \vec{a}\right| \left|\vec{b} \times \vec{a}\right|$

Solution

$\mathbf{a} \times [\mathbf{a} \times (\mathbf{a} \times \mathbf{b})] = \mathbf{a} \times (\mathbf{a} \cdot \mathbf{b} - \mathbf{a} \cdot \mathbf{b} \mathbf{b})$ $= (\mathbf{a} \cdot \mathbf{b}) (\mathbf{a} \times \mathbf{a}) - (\mathbf{a} \cdot \mathbf{a}) (\mathbf{a} \times \mathbf{b})$ $= (\mathbf{a} \cdot \mathbf{a}) (\mathbf{b} \times \mathbf{a}) \quad (\because \mathbf{a} \times \mathbf{a} = 0)$

Asked in: MHT CET Full Test 3

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