vectors scalar product help

Use properties of the scalar product to prove that for all vectors u,v $\displaystyle \in R^{2} $

$\displaystyle \lvert u+v \rvert^{2} + \lvert u-v \rvert^{2} = 2\lvert u \rvert^{2} + 2\lvert v \rvert^{2} $

i have expanded this got the above results, however I dont see how to use properties of the scalar product, i have simply applied some basic algebra?

Re: vectors scalar product help

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**Tweety** Use properties of the scalar product to prove that for all vectors u,v $\displaystyle \in R^{2} $

$\displaystyle \lvert u+v \rvert^{2} + \lvert u-v \rvert^{2} = 2\lvert u \rvert^{2} + 2\lvert v \rvert^{2} $

i have expanded this got the above results

I have no idea what that could mean!

$\displaystyle \|u\pm v\|^2=(u\pm v)\cdot(u\pm v)=u\cdot u\pm 2 u\cdot v+v\cdot v$

Re: vectors scalar product help

Stricty speaking $\displaystyle |u- v|^2$ is the scalar product of the vector u- v with itself- that's one property of "scalar product". $\displaystyle |u- v|^2= (u- v)\cdot(u- v)= u\cdot u- u\cdot v- v\cdot u+ v\cdot v$. Another property of the scalar product is that it is **commutative**- $\displaystyle u\cdot v= v\cdot u$ so that $\displaystyle -u\cdot v- v\cdot u= -u\cdot v- u\cdot v= -2u\cdot v$.

Do the same with $\displaystyle |u+ v|^2$.