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JEE Advanced2009MathematicsVector AlgebraActual

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

Options

  1. Aa , b , c are non-coplanar
  2. Bb , d are non-parallel
  3. Ca , b , d are non-coplanar
  4. Da, d are parallel and b , c are parallel

Correct answer

C. a , b , d are non-coplanar

Step-by-step solution

Let angle between a and b be ₁, c and d be ₂ and a b and c d be . Since, ( a b ) ( c d )=1 ₁ ₂ =1 ₁=90^ , ₂=90^ , =0^ a b , c d ,( a b ) |( c d ) So, a b =k( c d ) and a b =k( c d ) ( a b ) c =k( c d ) c and ( a b ) d =k( c d ) d [ array lll a & b & c array ]=0 and [ array lll a & b & d array ]=0 a , b , c and a , b , d are coplanar vectors, so options (a) and (b) are incorrect. Let b | d b = d As ( a b ) ( c d )=1( a b ) ( c b )= 1 [ a b c b ]= 1 [ c b a b ]= 1 c [ b ( a b )]= 1 c [ a -( b a ) b ]= 1 c a = 1 [ a b

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