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Homework Help: Orthogonal unit vectors also unit vectors?

  1. Mar 6, 2007 #1
    If two vectors v, w are both unit vectors, then v+w and v-w will be orthogonal, but are v+w and v-w also unit vectors?

    I would say no because the inner product of the two added, and the two subtracted would also have to be orthogonal.

    <(v+w)+(v-w),(v+w)-(v-w)>

    = <2v,2w>

    and <2v,2w> must be greater than zero for the product to be defined in the first place.

    *EDIT:
    If it helps, the way I originally showed that they were orthogonal was to take
    <v+w,v-w> = ||v||^2 - ||w||^2

    if they are unit vectors then ||v|| = 1 and ||w|| = 1 so

    <v+w,v-w> = 1 - 1 = 0 = orthogonal
     
    Last edited: Mar 6, 2007
  2. jcsd
  3. Mar 7, 2007 #2

    dextercioby

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    What is [itex] \left|| u-v\right|| ^2 [/itex] equal to ?
     
  4. Mar 7, 2007 #3

    HallsofIvy

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    ??? Are you saying that an inner product can't be negative? In any case, I see no reason for looking at <(v+w)+ (v-w),(v+w)-(v-w)> . The question was asking about v+w and v-w separately. You should be looking at
    ||v+w||2= <v+w,v+w> and ||v-w||2= <v-w,v-w>.

    Even more simply, what if v= w?

    *EDIT:
    If it helps, the way I originally showed that they were orthogonal was to take
    <v+w,v-w> = ||v||^2 - ||w||^2

    if they are unit vectors then ||v|| = 1 and ||w|| = 1 so

    <v+w,v-w> = 1 - 1 = 0 = orthogonal[/QUOTE]
     
  5. Mar 7, 2007 #4
    Yes because if v = w, then <2v,2v> must be positive because 4<v,v> must obey positivity.

    edit: whoops, I forgot the change the w at the end of my first post to v. This will work now, correct?
     
    Last edited: Mar 7, 2007
  6. Mar 7, 2007 #5
    I suppose that in the end dex's way would be the easiest and most straightforward (since it shows the norm is zero).
     
  7. Mar 7, 2007 #6

    Dick

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    ZERO!!????? What is zero?
     
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