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Vector question to think about

  1. Sep 16, 2007 #1
    my teacher told me to think about this and i don't seem to get it. given vectors X,Y,& Z; is there geometric significance when X*(YxZ)=0

    * is dot product and x is cross product
     
  2. jcsd
  3. Sep 16, 2007 #2

    cornfall

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    geometry

    Say X, Y and Z are orthonormal. Consider the part of X in the plane determined by the vectors Y and Z.
     
  4. Sep 17, 2007 #3

    HallsofIvy

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    The length of Y x Z is the area of a parallelogram with adjacent sides given by Y and Z.

    X*(YxZ) is the the volume of a parallopiped having sides, at one vertex, given by X, Y, and Z. You can see that by using the formulas [itex]X*Y= |X||Y|sin(\theta)[/itex] and length of [itex]X x Y= |X||Y|cos(\theta)[/itex].
     
  5. Sep 17, 2007 #4
    :mad: you shouldn't have just given him the answer
     
  6. Sep 17, 2007 #5

    robphy

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    While HallsofIvy gave the interpretation when that special product is generally nonzero, there's still some interpretation left to do [for the OP] for the zero case.
     
    Last edited: Sep 17, 2007
  7. Sep 17, 2007 #6

    isn't [itex]X x Y= |X||Y|sin(\theta)[/itex] and [itex]X*Y= |X||Y|cos(\theta)[/itex]?

    also, from that definition, would the angle between Y and Z equal to 0? when i break the given down, i get:
    |X||Y|cos(theta) x |X||Z|cos(theta).
    i cant really see the geometric significance.
     
  8. Sep 18, 2007 #7
    Combinations of cross products and dot products like that are known as triple products.
     
  9. Sep 18, 2007 #8
    wait so then if this triple product equals to 0, then does that mean the parallopiped is a cube?
     
  10. Sep 18, 2007 #9

    robphy

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    You are (mostly) correct.
    It is [itex] | \vec X \times \vec Y | = |\vec X| |\vec Y| |\sin\theta| [/itex] and [itex] \vec X \cdot \vec Y = |\vec X| |\vec Y| \cos\theta [/itex], where [itex]\theta[/tex] is the angle between the vectors. ([itex] \vec X \times \vec Y [/itex] is a vector with magnitude [itex] |\vec X| |\vec Y| |\sin\theta| [/itex] with direction perpendicular to the plane determined by [itex]\vec X [/itex] and [itex]\vec Y [/itex], according to the right-hand-rule.)
    (I suspect HallsofIvy's typo was due to a confusion over the symbols " * " and its synonym " X " for multiplication.)

    HallsofIvy gave the interpretation of the (scalar-)triple-product as the volume of a parallelopiped (a generally-slanted box with parallel sides) formed with those vectors. How would you describe this box if its volume were zero? What does that tell you about the relationship between [itex]\vec X[/itex], [itex]\vec Y[/itex] and [itex]\vec Z[/itex], along the lines of cornfall's suggestion?
     
  11. Sep 18, 2007 #10
    if X,Y, & Z are orthonormal, than is X = (YxZ)? also, if the volume of the box is zero, then its not a 3d figure, its 2d. in that case, its a square.
     
  12. Sep 18, 2007 #11

    robphy

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    So, what does that mean for vectors X, Y, and Z?
     
  13. Sep 18, 2007 #12
    is it that all three vectors are on the same plane?
     
    Last edited: Sep 18, 2007
  14. Sep 18, 2007 #13

    robphy

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    Ok that's one way that the triple product is zero.
    But suppose that X, Y, and Z are distinct nonzero vectors.
    In fact, take a special case when X, Y, and Z are all vectors of length 1.
    ( When X,Y,Z are mutually orthogonal, you have a cube... with volume 1. )

    Can you form a different parallelepiped with distinct nonzero vectors (with length 1) with a volume that is almost zero?... from there nudge things so that the volume is zero. What can you say about the vectors X,Y, and Z in that case? Now generalize to the general case.
     
  15. Sep 18, 2007 #14

    robphy

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    ah... you changed your answer on me.

    That's correct.
     
  16. Sep 18, 2007 #15
    well i spent a lot of time thinking about it, and it finally clicked. Thanks for all the help!
     
  17. Sep 19, 2007 #16

    HallsofIvy

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    Yes to the last- that's exactly what I said. No to the first. X x Y is a vector, not a number and what you give is its length.

    What you wrote makes no sense- you cannot take the the cross product of two numbers!
    What does X*(YxZ)= 0 tell you about X and YxZ? What does that tell you, then, about X and both Y and Z?
     
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