Question
Prove that $(\bar{a} \times \bar{b}) \cdot(\bar{c} \times \bar{d})\left|\begin{array}{ll}\bar{a} \cdot \bar{c} & \bar{b} \cdot \bar{c} \\ \bar{a} \cdot \bar{d} & \bar{b} \cdot \bar{d}\end{array}\right|$
$\mathrm{LHS}=(\bar{a} \times \bar{b}) \cdot(\bar{c} \times \bar{d})$
$=\bar{m} \cdot(\bar{c} \times \bar{d})$
$=(\bar{m} \times \bar{c}) \cdot \bar{d}$
$=[(\bar{a} \times \bar{b}) \times \bar{c}] \cdot \bar{d}$
$=[(\bar{c} \cdot \bar{a}) \bar{b}-(\bar{c} \cdot \bar{b}) \bar{a}] \cdot \bar{d}$
$=(\bar{c} \cdot \bar{a})(\bar{b} \cdot \bar{d})-(\bar{c} \cdot \bar{b})(\bar{a} \cdot \bar{d})$
$\begin{aligned} & =\left|\begin{array}{ll}\bar{c} \cdot \bar{a} & \bar{c} \cdot \bar{b} \\ a \cdot d & \frac{b}{b} \cdot \frac{d}{}\end{array}\right| \\ & =\left|\begin{array}{ll}\bar{a} \cdot \bar{c} & \bar{b} \cdot \bar{c} \\ \bar{a} \cdot d & \bar{b} \cdot d\end{array}\right|\end{aligned}$
[Dot product is commutative]
$=$ RHS.
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