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Let a = 2 i + x j + k and b = i - 2 j + y k be two orthogonal vectors. If the projection of a b on the vector c = i + 2 j - 2 k is 4 , then the value of | a b | is equal to

Options

  1. A18
  2. B9
  3. C81
  4. D6

Correct answer

B. 9

Step-by-step solution

Since a and b are orthogonal, a b = 0 . (2)(1) + (x)(-2) + (1)(y) = 0 y = 2x - 2 Now, computing the cross product: a b = vmatrix i & j & k 2 & x & 1 1 & -2 & y vmatrix = (xy + 2) i - (2y - 1) j + (-4 - x) k The projection of a b on c = i + 2 j - 2 k is given as 4 . ( a b ) c | c | = 4 Here, | c | = 1^2 + 2^2 + (-2)^2 = 3 . (xy + 2)(1) - (2y - 1)(2) + (-4 - x)(-2) 3 = 4 xy + 2 - 4y + 2 + 8 + 2x = 12 xy + 2x - 4y = 0 Substituting y = 2x - 2 into the equation: x(2x - 2) + 2x - 4(2x - 2) = 0 2x^2 - 2x + 2x - 8x + 8 = 0

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