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JEE MainMathematicsVector Algebra

Let a = 2 i + 2 j + k and b = 3 i + 4 k . Let v be a vector in the plane of a and b such that v is perpendicular to a . If the magnitude of v is equal to the area of the parallelogram formed by the vectors a and b , and the i component of v is positive, then v is equal to:

Options

  1. A1 15 5 (7 i - 20 j + 26 k )
  2. B4 i + 10 j - 3 k
  3. C8 i - 5 j - 6 k
  4. D1 3 (7 i - 20 j + 26 k )

Correct answer

D. 1 3 (7 i - 20 j + 26 k )

Step-by-step solution

The area of the parallelogram formed by a and b is given by | a b | . a b = vmatrix i & j & k 2 & 2 & 1 3 & 0 & 4 vmatrix = i (8 - 0) - j (8 - 3) + k (0 - 6) = 8 i - 5 j - 6 k Magnitude | a b | = 8^2 + (-5)^2 + (-6)^2 = 64 + 25 + 36 = 125 = 5 5 . Let the required vector be v = a + b since it lies in the plane of a and b . Given v a = 0 , we have: ( a + b ) a = 0 | a |^2 + ( a b ) = 0 We calculate: | a |^2 = 2^2 + 2^2 + 1^2 = 9 a b = (2)(3) + (2)(0) + (1)(4) = 10 So, 9 + 10 = 0 = -10k and = 9k for some scalar k . v

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