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