For any non - zero vectors $\bar{a}$ and $\overline{\mathrm{b}}…

For any non - zero vectors $\bar{a}$ and $\overline{\mathrm{b}},\left[\begin{array}{lll}\overline{\mathrm{b}} & \overline{\mathrm{a}} \times \overline{\mathrm{b}} & \overline{\mathrm{a}}\end{array}\right]=$
  1. $|\overline{\mathrm{a}} \times \overline{\mathrm{b}}|$
  2. $|\overline{\mathrm{a}} \times \overline{\mathrm{b}}|^{2}$
  3. 0
  4. $\overline{\mathrm{a}} \times \overline{\mathrm{b}}$

Solution

$\left[\begin{array}{lll}\bar{b} & \bar{a} \times \bar{b} & \bar{a}\end{array}\right]=\bar{b} \cdot[(\bar{a} \times \bar{b}) \times \bar{a}]$ $=-\bar{b}\left[\begin{array}{l}\bar{a} \times(\bar{a} \times \bar{b})\end{array}\right]$ $=-\bar{b}[(\mathrm{a} \cdot \mathrm{b}) \cdot \mathrm{a}-(\mathrm{a} \cdot \mathrm{a}) \cdot \mathrm{b}]=0$

Asked in: MHT CET 2020 (16 Oct Shift 2)

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