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Math Help - Vectors proof question

  1. #1
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    Vectors proof question

    Hi,can someone please explain how to solve this question to me.


    1) For any two vectors u and v, prove that |u + v|^2 + |u -v|^2 = 2(|u|^2 + |v|^2)

    In proving this, what geometric fact have your proved?

    Thanks heaps!!
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  2. #2
    Super Member ILikeSerena's Avatar
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    Re: Vectors proof question

    Quote Originally Posted by Vishak View Post
    Hi,can someone please explain how to solve this question to me.


    1) For any two vectors u and v, prove that |u + v|^2 + |u -v|^2 = 2(|u|^2 + |v|^2)

    In proving this, what geometric fact have your proved?

    Thanks heaps!!
    Hi Vishak!

    Note that |u+v|^2 = \langle u+v, u+v \rangle.
    Can you simplify that?
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  3. #3
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    Re: Vectors proof question

    I'm not sure what the triangle brackets mean, but ill guess - is it just u^2 + 2uv + v^2?
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    Re: Vectors proof question

    Quote Originally Posted by Vishak View Post
    1) For any two vectors u and v, prove that |u + v|^2 + |u -v|^2 = 2(|u|^2 + |v|^2)
    In proving this, what geometric fact have your proved?
    Surely you must know that \|u+v\|^2=(u+v)\cdot(u+v)~\&~\|u-v\|^2=(u-v)\cdot(u-v)~!

    Moreover, if vectors u~\&~v are adjacent sides of a parallelogram then (u+v)~\&~(u-v) are its diagonals.
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    Re: Vectors proof question

    Thanks guys, what about the second part - "what geometric fact have you proved?"
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  6. #6
    Super Member ILikeSerena's Avatar
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    Re: Vectors proof question

    Quote Originally Posted by Vishak View Post
    I'm not sure what the triangle brackets mean, but ill guess - is it just u^2 + 2uv + v^2?
    The triangle brackets are one of the ways you can write a dot product of vectors.
    You can also write it as (u+v) \cdot (u+v).

    And yes, that is what it is.

    Quote Originally Posted by Vishak View Post
    Thanks guys, what about the second part - "what geometric fact have you proved?"
    Care to guess now that you know that u~\&~v are adjacent sides of a parallelogram and (u+v)~\&~(u-v) are its diagonals?
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