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Math Help - is this correct?? - vectors

  1. #1
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    is this correct?? - vectors

    ok the ques is -
    The points A,B,C in R^3 are position vectors
    a=(1,1,2) b=(2,-1,4) c=(2,1,5)
    a) write down the vectors AB and AC
    b) Hence or otherwise find the position vector of the point D such that ABCD is a parallelogram.

    solution -
    i solved for
    a) AB=(1,-2,2) and AC=(1,0,3)
    b) ok the answer differs from my answer but this is what i said -
    using the definition of vector addition,
    a+b= c
    AB=-DC=-(1,-2,2)
    AC=(1,0,3)
    DC + AD = AC
    -(1,-2,3) + (x,y,z) =(1,0,3)
    therefore,
    (x,y,z) = (2,-2,5)

    am i incorrect for ques (b)??
    the answer was different!
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  2. #2
    MHF Contributor red_dog's Avatar
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    Let d=(x,y,z)

    If M is the point of intersection of diagonals AC and BD, then M is the midpoint of AC and BD.

    Then x_M=\frac{x_A+x_C}{2}=\frac{x_B+x_D}{2}\Rightarrow x_D=1

    y_M=\frac{y_A+y_C}{2}=\frac{y_B+y_D}{2}\Rightarrow y_D=2

    z_M=\frac{z_A+z_C}{2}=\frac{z_B+z_D}{2}\Rightarrow z_D=3

    \Rightarrow d=(1,2,3)
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