Q73 / 150 · MHT CET 2008 · Mathematics

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

  • A
    $(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{a}}) \cdot(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}})$
  • B
    $\overrightarrow{\mathbf{a}} \cdot(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}})-\overrightarrow{\mathbf{b}}(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})$
  • C
    $[\overrightarrow{\mathbf{a}} \cdot(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})] \overrightarrow{\mathbf{a}}$
  • Correct answer
    D
    $(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{a}})(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}})$

Solution

$\begin{aligned} \overrightarrow{\mathbf{a}} \times[\overrightarrow{\mathbf{a}}&\times(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})] \\ &=\overrightarrow{\mathbf{a}} \times\{(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}) \overrightarrow{\mathbf{a}}-(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{a}}) \overrightarrow{\mathbf{b}}\} \\ &=(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}})(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{a}})-(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{a}})(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}) \\ &=(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}) 0+(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{a}})(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}}) \\ &=(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{a}})(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}}) \end{aligned}$

Hence, the correct answer is option: 'D'.

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