Question

If $\alpha \overline{a}+\beta \overline{b}+\gamma \overline{c}=0$, then $(\overline{a}\times \overline{b})\times [(\overline{b}\times \overline{c})\times (\overline{c}\times \overline{a})]$ is equal to

• A. $\overline{0}$
• B. A vector $\perp$ plane of $\overline{a}$, $\overline{b}$ and $\overline{c}$
• C. A scalar quantity
• D. None of these

Answer 1

Answer: (A)

$\overline{0}$

Since $\alpha \overline{a}+\beta \overline{b}+\gamma \overline{c}=0$
$\therefore$ the vectors $\overline{a}$, $\overline{b}$ and $\overline{c}$ are coplanar
and thus $(\overline{c}\times \overline{b})$ and $(\overline{c}\times \overline{a})$ are coplanar.
$\therefore$ $(\overline{b}\times \overline{c})\times (\overline{c}\times \overline{a})=0$
Hence option $A$ becomes correct.