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    Vectors : Identities and Dot Products

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    How could you use the properties of the dot product to prove the following identities: (where u and v denote vectors in Rn)

    a) ||u + v||^2 + ||u-v||^2 = 2(||u||^2 + ||v||^2)
    b) ||u + v||^2 - ||u-v||^2 = 4u dot v

    dot = dot product
    ^ = power
    ||= distance.

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    Solution Preview

    ||u + v||^2= (u+v, u+v) where ( , ) shows the inner product (i.e. dot product). Then we have:

    ||u + v||^2= (u+v, u+v)= (u, u)+ (u, v)+ (v, u)+ (v, ...

    Solution Summary

    The properties of the dot product are used to prove the vector identities.