1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    User Avatar
    Science Advisor
    Homework Helper

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

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    ??? 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

    User Avatar
    Science Advisor
    Homework Helper

    ZERO!!????? What is zero?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Orthogonal unit vectors also unit vectors?
  1. Unit Vector (Replies: 1)

Loading...