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JEE MainMathematicsVector Algebra

Let a = i - j + 2 k and b = 2 i + j - k . If a vector c satisfies c a = b a and c ( i + 2 j + k ) = 8 , then | c |^2 is equal to:

Options

  1. A6
  2. B146
  3. C54
  4. D150

Correct answer

B. 146

Step-by-step solution

Given c a = b a . Rearranging the terms, we get: c a - b a = 0 ( c - b ) a = 0 This implies that the vector ( c - b ) is parallel to a . Therefore, for some scalar : c - b = a c = b + a We are given the dot product condition c ( i + 2 j + k ) = 8 . Let u = i + 2 j + k . Substituting c into the dot product: ( b + a ) u = 8 ( b u ) + ( a u ) = 8 Now, calculate a u and b u : a u = (1)(1) + (-1)(2) + (2)(1) = 1 - 2 + 2 = 1 b u = (2)(1) + (1)(2) + (-1)(1) = 2 + 2 - 1 = 3 Substitute these values back into the equation: 3

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