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JEE MainMathematicsVector Algebra

Let v = 2 i - j + 3 k and a = i + j + k . A vector b is coplanar with v and a , and is orthogonal to a . If the magnitude of b is 78 and b v > 0 , then the value of b v is equal to

Options

  1. A0
  2. B42
  3. C26
  4. D33

Correct answer

C. 26

Step-by-step solution

Since b is coplanar with v and a , let b = x v + y a . Given v = 2 i - j + 3 k and a = i + j + k . We have | v |^2 = 4 + 1 + 9 = 14 , | a |^2 = 1 + 1 + 1 = 3 , and v a = 2 - 1 + 3 = 4 . Since b is orthogonal to a , we have b a = 0 . (x v + y a ) a = x( v a ) + y| a |^2 = 0 4x + 3y = 0 y = - 4 3 x The magnitude of b is given as 78 , so | b |^2 = 78 . | b |^2 = (x v + y a ) (x v + y a ) = x^2| v |^2 + y^2| a |^2 + 2xy( v a ) 14x^2 + 3y^2 + 8xy = 78 Substituting y = - 4 3 x : 14x^2 + 3 (- 4 3 x )^2 + 8x (- 4 3 x ) = 7

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