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Just a quick verification

  1. Sep 1, 2005 #1
    If [tex] P , \; Q [/tex] and [tex] R [/tex] each represent a point in [tex] \mathbb{R} ^ 3 [/tex], then is it true that

    [tex] \left\| {\overrightarrow {PQ} \times \overrightarrow {PR} } \right\| = \left\| {\overrightarrow {PQ} \times \overrightarrow {QR} } \right\| \; {?} [/tex]
     
    Last edited: Sep 1, 2005
  2. jcsd
  3. Sep 1, 2005 #2

    quasar987

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    Almost any arbitrary choice of points is a counter exemple, the most obvious being with P = R = (0,0,0). If you meant "P,Q,R distincts", try it with P=(0,0,0), Q = (1,0,0), R = (0,3,0).
     
  4. Sep 1, 2005 #3
    Right, I meant distinct points P,Q, & R. :shy:
     
    Last edited: Sep 1, 2005
  5. Sep 1, 2005 #4
    PQ x PR = (1,0,0) x (0,3,0) = (0,0,3)
    PQ x QR = (1,0,0) x (-1,3,0) = (0,0,3)

    seem to have the same magnitude...
     
  6. Sep 1, 2005 #5
    It is always true. Furthermore, the resulting vectors are the same, not just their magnitude!
    Rewrite your vectors as

    [tex]\vec{A} \equiv \overrightarrow{PQ}[/tex],
    [tex]\vec{B} \equiv \overrightarrow{PR}[/tex], so that

    [tex](\vec{B} - \vec{A}) = \overrightarrow{QR}[/tex].

    Take the same cross products algebraically, keeping in mind that
    [tex]\vec{A} \times \vec{A} = 0[/tex], and that [tex]\vec{A} \times (\vec{B} - \vec{A}) = (\vec{A} \times \vec{B}) - (\vec{A} \times \vec{A})[/tex].
     
  7. Sep 1, 2005 #6

    quasar987

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    eeek!

    Sorry bomba.
     
    Last edited: Sep 1, 2005
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