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Math Help - Tetrahedron and the dot product

  1. #1
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    Tetrahedron and the dot product

    Here is the question
    "Two pairs of opposite edges of a tetrahedron are perpendicular. Prove that the third pair is also perpendicular."

    I let the 3 sides be a, b and c
    assume a is perpendicular to b
    assume b and c are perpendicular

    then a.b=0 and b.c=o

    need to show a.c=o

    but I am not sure how to proceed.

    Cheers
    Cabouli
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  2. #2
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    Opalg's Avatar
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    Quote Originally Posted by Cabouli View Post
    Here is the question
    "Two pairs of opposite edges of a tetrahedron are perpendicular. Prove that the third pair is also perpendicular."

    I let the 3 sides be a, b and c
    assume a is perpendicular to b
    assume b and c are perpendicular

    then a.b=0 and b.c=o

    need to show a.c=o

    but I am not sure how to proceed.
    Suppose the tetrahedron is OABC, with one vertex at the origin, and the others at points given by the vectors a,\ b,\ c. You are told that a.(c-b) = b.(a-c) = 0, and you want to prove that c.(b-a) = 0. Just use linearity: a.(c-b) = a.c-a.b,\ldots.
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