JEE Advance - Mathematics (2009 - Paper 1 Offline - No. 4)
If $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c $$ and $$\overrightarrow d $$ are unit vectors such that $$(\overrightarrow a \times \overrightarrow b )\,.\,(\overrightarrow c \times \overrightarrow d ) = 1$$ and $$\overrightarrow a \,.\,\overrightarrow c = {1 \over 2}$$, then
$$\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c $$ are non-coplanar
$$\overrightarrow b \,,\,\overrightarrow c ,\overrightarrow d $$ are non-coplanar
$$\overrightarrow b \,,\overrightarrow d $$ are non-parallel
$$\overrightarrow a ,\overrightarrow d $$ parallel and $$\overrightarrow b ,\overrightarrow c $$ are parallel
Explanation
The given equation, $$(\overrightarrow a \times \overrightarrow b )\,.\,(\overrightarrow c \times \overrightarrow d ) = 1$$, is possible only when $$|\overrightarrow a \times \overrightarrow b | = |\overrightarrow c \times \overrightarrow d | = 1$$ and $$(\overrightarrow a \times \overrightarrow b )||(\overrightarrow c \times \overrightarrow d )$$.
Since $$\overrightarrow a \,.\,\overrightarrow c = 1/2$$ and $$\overrightarrow b ||\overrightarrow d $$, we get $$|\overrightarrow c \times \overrightarrow d | \ne 1$$; hence, we conclude that the vectors $$\overrightarrow b $$ and $$\overrightarrow d $$ are non-parallel.
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