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Math Help - Under what conditions will these sets be orthogonal sets?

  1. #1
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    Under what conditions will these sets be orthogonal sets?

    Suppose that \{u,v,w\} is an orthogonal set. Under what conditions will the set S_1=\{u+v,u-v,w\} be an orthogonal set?

    Repeat for S_2=\{u+v-w,u-v+w,v+w\} and S_3=\{u-v,u+v,w-u,u+v+w\}.
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  2. #2
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    Quote Originally Posted by Runty View Post
    Suppose that \{u,v,w\} is an orthogonal set. Under what conditions will the set S_1=\{u+v,u-v,w\} be an orthogonal set?

    Repeat for S_2=\{u+v-w,u-v+w,v+w\} and S_3=\{u-v,u+v,w-u,u+v+w\}.
    vectors x and y are orthogonal if: (x|y) = 0 ( scalar product )

    if set { u,v,w } is orthogonal that means that:

    ( u|v ) = 0 and ( u|w ) = 0 and ( v|w ) = 0

    what conditions do you need for set { u+v, u-v,w } to be orthogonal?

    it must be:

    ( u + v | u - v ) = 0 and ( u + v | w ) = 0 and ( u - v | w ) = 0

    1. ( u + v | u - v ) = ( u|u ) + ( v|u ) + ( u|v ) - ( v|v ) = ( u|u ) - ( v|v ) ---- so first condition is:

    ( u|u ) = ( v|v )

    2. it is ok ( u + v | w ) = ( u|w ) + ( v|w ) = 0
    3. it is ok

    so condition is ( u|u ) = ( v|v )

    S_2 and S_3 in the same way
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  3. #3
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    I've gone through all of it, and I'm somewhat concerned about the answer for S_3. I end up getting u\cdot u=v\cdot v,u\cdot u=w\cdot w,u\cdot u=0, so v\cdot v=w\cdot w=0. This would imply (I think) that S_3 is always orthogonal. Is this true, or did I do something wrong?
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