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JEE MainMathematicsVector Algebra

Let p and q be two vectors such that | p | = 2 , | q | = 3 , and | p + q |^2 = 13 . Let S be the set defined as S = r r p = q and | r | 1, 2, 3, 4 . The number of elements in the set S is

Options

  1. A0
  2. B3
  3. C6
  4. D8

Correct answer

C. 6

Step-by-step solution

First, we find the dot product of p and q using the given magnitude of their sum: | p + q |^2 = | p |^2 + | q |^2 + 2 p q 13 = 4 + 9 + 2 p q 2 p q = 0 p q = 0 Since p and q are orthogonal, the equation r p = q has solutions. The general solution represents a line in 3D space parallel to p . Taking the cross product with p on both sides: p ( r p ) = p q ( p p ) r - ( p r ) p = p q Let p r = . Then 4 r - p = p q , which gives r = p q 4 + 4 p . Let r ₀ = p q 4 . Since p q , | p q | = | p || q | = 6 . Thus, | r ₀| = 6

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