If a, b, c and d are unit vectors such that ( a b) ( c d) = 1 and a c = 1 2 , then

JEE Advanced 2009 Mathematics question

If , , and are unit vectors such that

( a× b)·( c× d) = 1 and a· c = ,

then

(A), , are non-coplanar
(B), , are non-coplanar
(C), are non-parallelCorrect
(D), are parallel and , are parallel

Answer: (C)

Solution

and , so the dot product equals 1 only if both cross products are unit vectors in the same direction: , , and the planes of and coincide with the same orientation. All four vectors

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Question ID VK-037696