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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

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

Solution Summary

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

Solution Preview

a)
||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, ...

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