If $(\mathrm{a} \times \mathrm{b}) \times \overline{\mathrm{c}}=\overline{\mathrm{a}}…

If $(\mathrm{a} \times \mathrm{b}) \times \overline{\mathrm{c}}=\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})$, where $\overline{\mathrm{a}}, \overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ are any three vectors such that $\overline{\mathrm{a}} \cdot \overline{\mathrm{b}} \neq 0$, $\overline{\mathrm{b}} \cdot \overline{\mathrm{c}} \neq 0$, then $\mathrm{a}$ and $\mathrm{c}$ are
  1. inclined at an angle of $\pi / 3$ between them
  2. inclined at an angle of $\pi / 6$ between them
  3. perpendicular
  4. parallel

Solution

$(\mathrm{a} \times \mathrm{b}) \times \overline{\mathrm{c}}=\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}}), \overline{\mathrm{a}} \cdot \overline{\mathrm{b}} \neq 0, \overline{\mathrm{b}} \cdot \overline{\mathrm{c}} \neq 0$ $\Rightarrow(\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}) \overline{\mathrm{b}}-(\overline{\mathrm{b}} \cdot \overline{\mathrm{c}}) \overline{\mathrm{a}}=(\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}) \overline{\mathrm{b}}-(\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}) \overline{\mathrm{c}}$ $(\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}) \overline{\mathrm{c}}=(\overline{\mathrm{b}} \cdot \overline{\mathrm{c}}) \overline{\mathrm{a}}$ $\overline{\mathrm{a}} \| \overline{\mathrm{c}}$

Asked in: JEE Main 2006

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