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JEE MainMathematicsVector Algebra

Let a = i + j + k , b = -2 i - j - k , and c = i + 2 j + k be three vectors. If a vector x satisfies x a = b a and x c = 0 , then which of the following statements is TRUE?

Options

  1. AThe magnitude of the projection of x on a is 1 4 3 and the direction of the projection vector is the same as t
  2. BThe magnitude of the projection of x on a is 1 4 and the direction of the projection vector is opposite to the
  3. CThe magnitude of the projection of x on a is 1 4 3 and the direction of the projection vector is opposite to t
  4. DThe magnitude of the projection of x on a is 1 12 and the direction of the projection vector is opposite to th

Correct answer

C. The magnitude of the projection of x on a is 1 4 3 and the direction of the projection vector is opposite to t

Step-by-step solution

Given x a = b a ( x - b ) a = 0 This implies that the vector ( x - b ) is collinear with a . Therefore, we can write: x - b = a for some scalar x = b + a We are also given that x c = 0 . Substituting the expression for x : ( b + a ) c = 0 b c + ( a c ) = 0 Now, let us calculate the required dot products: b c = (-2)(1) + (-1)(2) + (-1)(1) = -2 - 2 - 1 = -5 a c = (1)(1) + (1)(2) + (1)(1) = 1 + 2 + 1 = 4 Substitute these values into the equation: -5 + 4 = 0 = 5 4 Thus, x = b + 5 4 a . The scalar projection of x on a i

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