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ABCD forms a rectangle. With 3 points, A,B,C, find D.

  1. Jan 31, 2016 #1
    1. The problem statement, all variables and given/known data
    Given A = [2, 9, 8], B = [6, 4, −2] and C = [7, 15, 7], show that AB and AC are perpendicular, then find D so that ABCD forms a rectangle.

    2. Relevant equations

    Dot Product

    3. The attempt at a solution
    The vector AB = B - A = [4,-5,-10]
    The vector AC = C - A = [5,6,-1]

    AB⋅AC = 0 if they are perpendicular

    (4*5) + (-5*6) + (-10*-1) = 20 - 30 + 10 = 0

    AB and AC are perpendicular.

    I'm not sure how to find the point D though. Seeing as it's a rectangle, the distance from CD = AB? And the distance from BD = AC? Can I just use Pythagoras to find the distance from A to D:

    AD2 = AB2+AC2

    AD2 = [4,-5,-10]2 + [5,6,-1]2

    AD2 = [41,61,101]

    I got to here and feel like I might have over-thought the problem a bit...

    I've arranged the letters below to show how i'm setting up the points in a rectangle

    BD
    AC
     
  2. jcsd
  3. Jan 31, 2016 #2

    mfb

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    You can work with distances, but there is a much easier approach. What do you know about vectors of opposite sides in a rectangle, e.g, DC and AB?
     
  4. Jan 31, 2016 #3
    They have the same length.
     
  5. Jan 31, 2016 #4

    mfb

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    You can make a stronger statement (which is true even in general parallelograms, and gave them their name).
     
  6. Jan 31, 2016 #5
    They are parallel, so one will be a multiple of the other?
     
  7. Jan 31, 2016 #6

    mfb

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    They are parallel and have the same length. What does that make together?
     
  8. Jan 31, 2016 #7
    CD = AB = [4,-5,-10]

    So if we start from C = [7, 15, 7] we just add the vector to that to get the coordinates of D?

    D (coordinates) = C+ AB = [7, 15, 7] + [4,-5,-10] = [11, 10,-3]
     
  9. Jan 31, 2016 #8

    mfb

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    Right.
     
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