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COMEDK2024Evening ShiftMathematicsThree Dimensional GeometryActual

The co-ordinate of the foot of the perpendicular from P(1,8,4) on the line joining R(0,-1,3) and Q(2,-3,-1) is

Options

  1. A( -5 3 , 2 3 , 19 3 )
  2. B( 5 3 , 2 3 , 19 3 )
  3. C( 5 3 , 2 3 , -19 3 )
  4. D( -5 3 , -2 3 , -19 3 )

Correct answer

A. ( -5 3 , 2 3 , 19 3 )

Step-by-step solution

The line passes through R(0, -1, 3) and Q(2, -3, -1) . The direction vector of the line is v = Q - R = (2 - 0, -3 - (-1), -1 - 3) = (2, -2, -4) . The equation of the line can be written in parametric form as x = 2t , y = -1 - 2t , z = 3 - 4t . Let M be the foot of the perpendicular from P(1, 8, 4) on the line. The coordinates of M are (2t, -1 - 2t, 3 - 4t) . The vector PM = (2t - 1, -1 - 2t - 8, 3 - 4t - 4) = (2t - 1, -2t - 9, -4t - 1) . Since PM is perpendicular to the line direction v = (2, -2, -4) , their dot pr

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