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MHT CET202523 Apr 2025Morning ShiftMathematicsVector AlgebraActual

Let a = i + j + k , b and c = j - k be three vectors such that a b = c and a c =0 . If the length of projection vector of the vector b on the vector a c is l , then the value of 3 l^2 is

Options

  1. A1
  2. B2
  3. C4
  4. D6

Correct answer

B. 2

Step-by-step solution

Given vectors: a = i + j + k c = j - k The condition a c = 0 is satisfied since a and c are orthogonal. Let d = a c . Using the determinant form: d = vmatrix i & j & k 1 & 1 & 1 0 & 1 & -1 vmatrix = -2 i + j + k The magnitude is | d | = (-2)^2 + 1^2 + 1^2 = 6 . The projection length of b on d is l = | b d | | d | . Using the scalar triple product property: b ( a c ) = -[ a b c ] = -( a b ) c Given a b = c , this becomes - c c = -| c |^2 . Since | c |^2 = 1^2 + (-1)^2 = 2 , we have b d = -2 . The projection length i

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