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MHT CET202523 Apr 2025Morning ShiftMathematicsVector AlgebraActual

Let a =2 i + j + k , b = i +2 j - k and vector c be coplanar. If c is perpendicular to a , then c is

Options

  1. A- i +2 k
  2. B- i + j + k
  3. Ci -2 j
  4. D- j + k

Correct answer

D. - j + k

Step-by-step solution

Given vectors a =2 i + j + k and b = i +2 j - k , find vector c = x i + y j + z k that is coplanar with a and b and perpendicular to a . Coplanarity requires c ( a b ) = 0 . Compute a b = vmatrix i & j & k 2 & 1 & 1 1 & 2 & -1 vmatrix = i (-1 - 2) - j (-2 - 1) + k (4 - 1) = -3 i + 3 j + 3 k . Dot product gives -3x + 3y + 3z = 0 ; dividing by 3 yields -x + y + z = 0 . Perpendicularity requires c a = 0 , giving 2x + y + z = 0 . Subtracting the first equation from the second: (2x + y + z) - (-x + y + z) = 0 3x = 0 x =

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