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If $\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d}$ are coplanar vectors, then $(\mathbf{a} \times \mathbf{b}) \times(\mathbf{c} \times \mathbf{d})=$
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$\begin{aligned}
& (\mathbf{a} \times \mathbf{b}) \times(\mathbf{c} \times \mathbf{d})=[\mathbf{a} \mathbf{b} \mathbf{d}] \mathbf{c}-[\mathbf{a} \mathbf{b} \mathbf{c}] \mathbf{d} \\
& \text { a, b, c, }, \mathbf{d} \text { are coplanar vectors } \\
& \therefore[\mathbf{a} \mathbf{b} \mathbf{d}]=[\mathbf{a} \mathbf{b} \mathbf{c}]=0 . \text { So, }(\mathbf{a} \times \mathbf{b}) \times(\mathbf{c} \times \mathbf{d})=\mathbf{0}
\end{aligned}$
So, $(\mathbf{a} \times \mathbf{b}) \times(\mathbf{c} \times \mathbf{d})=\mathbf{0}$
& (\mathbf{a} \times \mathbf{b}) \times(\mathbf{c} \times \mathbf{d})=[\mathbf{a} \mathbf{b} \mathbf{d}] \mathbf{c}-[\mathbf{a} \mathbf{b} \mathbf{c}] \mathbf{d} \\
& \text { a, b, c, }, \mathbf{d} \text { are coplanar vectors } \\
& \therefore[\mathbf{a} \mathbf{b} \mathbf{d}]=[\mathbf{a} \mathbf{b} \mathbf{c}]=0 . \text { So, }(\mathbf{a} \times \mathbf{b}) \times(\mathbf{c} \times \mathbf{d})=\mathbf{0}
\end{aligned}$
So, $(\mathbf{a} \times \mathbf{b}) \times(\mathbf{c} \times \mathbf{d})=\mathbf{0}$
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