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JEE MainMathematicsVector Algebra

Let p , q , and r be three unit vectors such that p q + q r + r p = - 3 2 . The value of 4 | p q + q r + r p |^2 is equal to

Options

  1. A9
  2. B0
  3. C36
  4. D27

Correct answer

D. 27

Step-by-step solution

We know that | p + q + r |^2 = | p |^2 + | q |^2 + | r |^2 + 2( p q + q r + r p ) . Since p , q , and r are unit vectors, | p |^2 = | q |^2 = | r |^2 = 1 . Substituting the given values, we get: | p + q + r |^2 = 1 + 1 + 1 + 2 (- 3 2 ) = 3 - 3 = 0 . This implies that p + q + r = 0 . Taking the cross product of this equation with p , we get: p ( p + q + r ) = 0 p p + p q + p r = 0 0 + p q - r p = 0 p q = r p . Similarly, taking the cross product with q yields q r = p q . Therefore, p q + q r + r p = 3( p q ) . The e

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