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JEE MainMathematicsVector Algebra

Let a = 2 i + 2 j - k and b = 4 i + 2 j + 3 k . If a vector v satisfies v a = b a and v a = 0 , then the value of | v |^2 is

Options

  1. A56
  2. B20
  3. C29
  4. D101

Correct answer

B. 20

Step-by-step solution

Given v a = b a ( v - b ) a = 0 This implies that v - b is parallel to a , so we can write: v - b = a v = b + a We are also given that v a = 0 . Taking the dot product of v with a : ( b + a ) a = 0 b a + | a |^2 = 0 Now, calculate b a and | a |^2 : a = 2 i + 2 j - k b = 4 i + 2 j + 3 k b a = (4)(2) + (2)(2) + (3)(-1) = 8 + 4 - 3 = 9 | a |^2 = 2^2 + 2^2 + (-1)^2 = 4 + 4 + 1 = 9 Substituting these values into the equation: 9 + (9) = 0 = -1 Therefore, v = b - a v = (4 i + 2 j + 3 k ) - (2 i + 2 j - k ) = 2 i + 0 j + 4

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