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AP EAMCET202319 May 2023Morning ShiftMathematicsThree Dimensional GeometryActual

Let A B C D be a parallelogram and 2 i + j , 4 i +5 j +4 k and - i -4 j -3 k be the position vectors of the vertices A, B , D respectively. Then the position vector of one of the point of trisection of the diagonal AC is

Options

  1. A1 3 (5 i +2 j - k )
  2. B1 3 (5 i +2 j + k )
  3. C1 3 (5 i +4 j + k )
  4. D1 3 (3 i +2 j + k )

Correct answer

A. 1 3 (5 i +2 j - k )

Step-by-step solution

Since diagonals of parallelogram bisect each other at mid point. Hence coordinates of M aligned & = ( -1+4 2 , -4+5 2 , -3+4 2 ) & = ( 3 2 , 1 2 , 1 2 ) aligned Since M is also mid point of point A and C Hence, ( 2+x 2 , 1+y 2 , 0+z 2 )= ( 3 2 , 1 2 , 1 2 ) aligned & 2+x 2 = 3 2 & x=1 aligned | array l|l 1+y 2 = 1 2 & z 2 = 1 2 z=0 array | Hence coordinate of c is c(1,0,1) Since a trisection divides a line in 1: 2 and 2: 1 ratio. When 1: 2 is in consideration then coordinates of T₁ aligned & =T₁ ( 1(1)+2(2) 1+2 , 1

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