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JEE MainMathematicsVector Algebra

Let a = i + j + k and b = i - j + k . Let c be a vector satisfying ( a - 2 b ) c = c (-2 a - b ) . If the area of the parallelogram with adjacent sides c and a is 12 2 , then | c |^2 is equal to :

Options

  1. A144
  2. B864
  3. C1296
  4. D72

Correct answer

A. 144

Step-by-step solution

Given ( a - 2 b ) c = c (-2 a - b ) Using the anti-commutative property of the cross product: ( a - 2 b ) c = -(-2 a - b ) c ( a - 2 b ) c = (2 a + b ) c Bringing all terms to one side: ( a - 2 b ) c - (2 a + b ) c = 0 (- a - 3 b ) c = 0 This implies that c is collinear with ( a + 3 b ) , so: c = ( a + 3 b ) The area of the parallelogram with adjacent sides c and a is given by | c a | : c a = ( a + 3 b ) a = 3 ( b a ) Calculate b a : b a = ( i - j + k ) ( i + j + k ) b a = i (-1 - 1) - j (1 - 1) + k (1 - (-1)) = -2

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