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Prove that $(\overrightarrow{a}\times\overrightarrow{b}) . (\overrightarrow{c}\times\overrightarrow{d}) + (\overrightarrow{b}\times\overrightarrow{c}) . (\overrightarrow{a}\times\overrightarrow{d}) + (\overrightarrow{c}\times\overrightarrow{a}) . (\overrightarrow{b}\times\overrightarrow{d})=0$

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  • Scalar product of four vectors $ ( \overrightarrow a \times \overrightarrow b).(\overrightarrow c \times \overrightarrow a) = \begin{vmatrix} \overrightarrow a.\overrightarrow c & \overrightarrow a.\overrightarrow d \\ \overrightarrow b.\overrightarrow c & \overrightarrow b.\overrightarrow d \end{vmatrix}$
LHS = $ \begin{vmatrix} \overrightarrow a.\overrightarrow c & \overrightarrow a.\overrightarrow d \\ \overrightarrow b.\overrightarrow c & \overrightarrow b.\overrightarrow d \end{vmatrix} + \begin{vmatrix} \overrightarrow b.\overrightarrow a & \overrightarrow b.\overrightarrow d \\ \overrightarrow c.\overrightarrow a & \overrightarrow c.\overrightarrow d \end{vmatrix} + \begin{vmatrix} \overrightarrow c.\overrightarrow b & \overrightarrow c.\overrightarrow d \\ \overrightarrow a.\overrightarrow b & \overrightarrow a.\overrightarrow d \end{vmatrix}$
$ = (\overrightarrow a.\overrightarrow c)(\overrightarrow b.\overrightarrow d)-(\overrightarrow b.\overrightarrow c)(\overrightarrow a.\overrightarrow d)+(\overrightarrow b.\overrightarrow a)(\overrightarrow c.\overrightarrow d)$
$ -(\overrightarrow c.\overrightarrow a)(\overrightarrow b.\overrightarrow d)+(\overrightarrow c.\overrightarrow b)(\overrightarrow a.\overrightarrow d)-(\overrightarrow a.\overrightarrow b)(\overrightarrow c.\overrightarrow d)$
= 0 = RHS

 

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