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JEE Main202621 January 2026Morning ShiftMathematicsThree Dimensional GeometryActual

Let ( , , ) be the co-ordinates of the foot of the perpendicular drawn from the point (5,4,2) on the line r =(- i +3 j + k )+ (2 i +3 j - k ) . Then the length of the projection of the vector i + j + k on the vector 6 i +2 j +3 k is :

Options

  1. A15 7
  2. B4
  3. C18 7
  4. D3

Correct answer

C. 18 7

Step-by-step solution

Line: r = (-1, 3, 1) + (2, 3, -1) , Point P = (5, 4, 2). Any point on line: Q = (-1 + 2 , 3 + 3 , 1 - ) . PQ = (-6 + 2 , -1 + 3 , -1 - ) . For perpendicular: PQ (2, 3, -1) = 0 gives 14 - 14 = 0 , so = 1 . Foot ( , , ) = (1, 6, 0) . Projection of (1, 6, 0) on (6, 2, 3) : |6 + 12 + 0| 49 = 18 7 .

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