JEE Advance - Mathematics (2004 - No. 3)
If $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c $$ and $$\overrightarrow d $$ are distinct vectors such that
$$\,\overrightarrow a \times \overrightarrow c = \overrightarrow b \times \overrightarrow d $$ and $$\overrightarrow a \times \overrightarrow b = \overrightarrow c \times \overrightarrow d \,.$$ Prove that
$$\left( {\overrightarrow a - \overrightarrow d } \right).\left( {\overrightarrow b - \overrightarrow c } \right) \ne 0\,\,i.e.\,\,\,\overrightarrow a .\overrightarrow b + \overrightarrow d .\overrightarrow c \ne \overrightarrow d .\overrightarrow b + \overrightarrow a .\overrightarrow c $$
$$\,\overrightarrow a \times \overrightarrow c = \overrightarrow b \times \overrightarrow d $$ and $$\overrightarrow a \times \overrightarrow b = \overrightarrow c \times \overrightarrow d \,.$$ Prove that
$$\left( {\overrightarrow a - \overrightarrow d } \right).\left( {\overrightarrow b - \overrightarrow c } \right) \ne 0\,\,i.e.\,\,\,\overrightarrow a .\overrightarrow b + \overrightarrow d .\overrightarrow c \ne \overrightarrow d .\overrightarrow b + \overrightarrow a .\overrightarrow c $$
$$\overrightarrow a - \overrightarrow d $$ is perpendicular to $$\overrightarrow b - \overrightarrow c $$
$$\overrightarrow a - \overrightarrow d $$ is parallel to $$\overrightarrow b - \overrightarrow c $$
$$\left( {\overrightarrow a - \overrightarrow d } \right).\left( {\overrightarrow b - \overrightarrow c } \right) = 0$$
$$\overrightarrow a .\overrightarrow b + \overrightarrow d .\overrightarrow c = \overrightarrow d .\overrightarrow b + \overrightarrow a .\overrightarrow c $$
The vectors are coplanar
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