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Vector Geometry

  1. Aug 13, 2010 #1
    Could someone please give me a hint on this question?

    In the tetrahedron ABCD, AB is perpendicular to DC and AD is perpendicular to BC, prove that DB is perpendicular to AC.

    This is what I am stuck on:

    DB.AC = (DC+CB).(AD+dc)
    =DC.AD +DC.DC+CB.AD+CB.DC
    =(CA+AD).AD+d.d+(CA+AB).DC
    =CA.AD+AD.AD+d.d+CA.DC+ab.dc
    =-c.d+d.d+d.d+-c.(CA+AD)
    -c.d+d.d+d.d+c.c+-c.d
    =2d.d+c.c-2(c.d)

    See attachment.
     

    Attached Files:

  2. jcsd
  3. Aug 17, 2010 #2
    Are you sure you wrote down the question correctly? As it stands, I don't think any tetrahedron's going to satisfy your givens. It seems to me that both AB and BC would have to be perpendicular to the plane through ABC; but then AB and BC would parallel, contradicting the fact that they intersect. However, it's 3:45am by me, so I might have missed something.
     
  4. Aug 17, 2010 #3

    ehild

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    Gold Member

    The problem is OK.
    If you project the vertex A onto the plane BCD the projected edges of the tetrahedron are also perpendicular. This might help...

    ehild
     
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