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JEE Advanced2021MathematicsVector AlgebraActual

Let O be the origin and O A → = 2 i ^ + 2 j ^ + k ^ , O B → = i ^ - 2 j ^ + 2 k ^ and O C → = 1 2 O B → - λ O A → for some λ > 0 . If O B → × O C → = 9 2 , then which of the following statements is (are) TRUE?

Options

  1. AProjection of O C → and O A → is - 3 2
  2. BArea of the triangle O A B is 9 2
  3. CArea of the triangle A B C is 9 2
  4. DThe acute angle between the diagonals of the parallelogram with adjacent sides O A → and O C → is

Correct answer

A. Projection of O C → and O A → is - 3 2

Step-by-step solution

O A → = 2 i ^ + 2 j ^ + k ^ O B → = i ^ − 2 j ^ + 2 k ^ O C → = 1 2 O B → − λ O A → ; λ > 0 O B → × O C → = 9 2 ⇒ O B → × 1 2 O B → − λ O A → = 9 2 ⇒ 1 2 O B → × O B → − λ O A → = 9 2 ⇒ 1 2 λ O A → × O B → = 9 2 ⇒ λ 2 O A → × O B → = 9 2             . . . 1 now, O A → ×

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