The vectors $$\overrightarrow a ,\,\overrightarrow b ,\,\overrightarrow c $$   and $$\overrightarrow d $$ are such that $$\overrightarrow a \times \overrightarrow b = \overrightarrow c \times \overrightarrow d $$    and $$\overrightarrow a \times \overrightarrow c = \overrightarrow b \times \overrightarrow d .$$    Which of the following is/are correct ?
$$\eqalign{ & 1.\,\left( {\overrightarrow a - \overrightarrow d } \right) \times \left( {\overrightarrow b - \overrightarrow c } \right) = \overrightarrow 0 \cr & 2.\,\left( {\overrightarrow a \times \overrightarrow b } \right) \times \left( {\overrightarrow c \times \overrightarrow d } \right) = \overrightarrow 0 \cr} $$
Select the correct answer using the code given below :

Solution :
$$\eqalign{ & \left( {\overrightarrow a - \overrightarrow d } \right) \times \left( {\overrightarrow b - \overrightarrow c } \right) \cr & = \overrightarrow a \times \overrightarrow b - \overrightarrow d \times \overrightarrow b - \overrightarrow a \times \overrightarrow c + \overrightarrow d \times \overrightarrow c \cr & = \overrightarrow c \times \overrightarrow d - \overrightarrow d \times \overrightarrow b - \overrightarrow b \times \overrightarrow d - \overrightarrow c \times \overrightarrow d \cr & = - \overrightarrow d \times \overrightarrow b + \overrightarrow d \times \overrightarrow b \cr & = 0 \cr & {\text{Again }}\left( {\overrightarrow a \times \overrightarrow b } \right) = \left( {\overrightarrow c \times \overrightarrow d } \right){\text{ given}} \cr & \Rightarrow \left( {\overrightarrow a \times \overrightarrow b } \right) \times \left( {\overrightarrow c \times \overrightarrow d } \right) = \left( {\overrightarrow c \times \overrightarrow d } \right) \times \left( {\overrightarrow c \times \overrightarrow d } \right) = 0\,\,\,\,\,\left( {{\text{as }}\overrightarrow a \times \overrightarrow a = 0} \right) \cr} $$
So both $$\left( 1 \right)$$ and $$\left( 2 \right)$$ are correct.