Question

If $[\overline{a}\times \overline{b}\overline{b}\times \overline{c}\overline{c}\times \overline{a}]=\lambda [\overline{a}\overline{b}\overline{c}{]}^2$, then $\lambda =$

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is B $1$

$[\overline{a}\times \overline{b}\overline{b}\times \overline{c}\overline{c}\times \overline{a}]$
$=(\overline{a}\times \overline{b}).\left[\right(\overline{b}\times \overline{c})\times (\overline{c}\times \overline{a}\left)\right]$

$=(\overline{a}\times \overline{b}).\left[\right(\overline{b}\times \overline{c}).\overline{a}\overline{c}-(\overline{b}\times \overline{c}).\overline{c}\overline{a}]$ using { $(\overline{a}\times \overline{b})\times \overline{c}=(\overline{a}.\overline{c})b-(\overline{b}.\overline{c})a$ }
$=(\overline{a}\times \overline{b}).[\overline{a}.(\overline{b}\times \overline{c})\overline{c}-0]$ as $(\overline{b}\times \overline{c}).\overline{c}=\left[\overline{b}\overline{c}\overline{c}\right]=0$

$=[\overline{a}\overline{b}\overline{c}{]}^2$

$\Rightarrow \lambda =1$