Question

If $(\overline{a}\times \overline{b})\times (\overline{c}\times \overline{d})=\lambda \overline{c}+\mu \overline{d}$ then $\lambda ,\mu =$

• A. $\left[\overline{a}\overline{b}\overline{d}\right],-\left[\overline{a}\overline{b}\overline{c}\right]$
• B. $\left[\overline{a}\overline{b}\overline{c}\right],\left[\overline{b}\overline{c}\overline{d}\right]$
• C. $\left[\overline{a}\overline{b}\overline{d}\right],\left[\overline{b}\overline{a}\overline{c}\right]$
• D. $1,0$

Answer 1

The correct option is A $\left[\overline{a}\overline{b}\overline{d}\right],-\left[\overline{a}\overline{b}\overline{c}\right]$

$(\overline{a}\times \overline{b})\times (\overline{c}\times \overline{d})$
$=\overline{c}\left(\right(\overline{a}\times \overline{b}).\overline{d})-\overline{d}(\overline{a}\times \overline{b}).\left(\overline{c}\right)$
$=\overline{c}\left(\right[abd\left]\right)-\overline{d}\left(\right[abc\left]\right)$
$=\lambda \overline{c}+\overline{d}\mu$
Hence by comparing coefficients we get,
$\lambda =\left(\right[abd\left]\right)$ and $\mu =(-[abc\left]\right)$