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COMEDK2024Morning ShiftMathematicsThree Dimensional GeometryActual

The foot of the perpendicular from (2,4,-1) to the line x+5= 1 4 (y+3)=- 1 9 (z-6) is

Options

  1. A(-4,-1,-3)
  2. B(-4,1,-3)
  3. C(4,-1,-3)
  4. D(-4,-1,3)

Correct answer

B. (-4,1,-3)

Step-by-step solution

The given line is x+5 = y+3 4 = z-6 -9 = . Any point on the line is P( - 5, 4 - 3, -9 + 6) . Let A = (2, 4, -1) . The direction vector of the line is v = (1, 4, -9) . The vector AP = ( - 5 - 2, 4 - 3 - 4, -9 + 6 - (-1)) = ( - 7, 4 - 7, -9 + 7) . Since AP is perpendicular to the line, AP v = 0 . 1( - 7) + 4(4 - 7) - 9(-9 + 7) = 0 . - 7 + 16 - 28 + 81 - 63 = 0 . 98 - 98 = 0 = 1 . Substituting = 1 into the coordinates of P : x = 1 - 5 = -4 , y = 4(1) - 3 = 1 , z = -9(1) + 6 = -3 . The foot of the perpendicular is (-4,

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