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JEE Main202226 Jun 2022Evening ShiftMathematicsVector AlgebraActual

Let a → = i ^ + j ^ + 2 k ^ , b → = 2 i ^ - 3 j ^ + k ^ and c → = i ^ - j ^ + k ^ be the three given vectors. Let v → be a vector in the plane of a → and b → whose projection on c → is 2 3 . If v → , j ^ = 7 , then v → · i ^ + k ^ is equal to

Options

  1. A6
  2. B7
  3. C8
  4. D9

Correct answer

D. 9

Step-by-step solution

Since vector v → ,   a →   &   b → are coplanar then, v → = λ 1 a → + λ 2 b → , where λ 1 , λ 2 ∈ ℝ . ⇒ v → = λ 1 + 2 λ 2 i ^ + λ 1 - 3 λ 2 j ^ + 2 λ 1 + λ 2 k ^ ∵ Projection of v → on c → is 2 3 ∴ λ 1 + 2 λ 2 - λ 1 + 3 λ 2 + 2 λ 1 + λ 2 3 = 2 3 ∴     λ 1 + 3 λ 2 = 1             .

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