vectors scalar product help

**Tweety** · November 14th 2012, 11:55 PM

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

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

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?

**Plato** · November 15th 2012, 03:46 AM

Originally Posted by Tweety

Use properties of the scalar product to prove that for all vectors u,v $\in R^{2}$
$\lvert u+v \rvert^{2} + \lvert u-v \rvert^{2} = 2\lvert u \rvert^{2} + 2\lvert v \rvert^{2}$
i have expanded this got the above results

I have no idea what that could mean!

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

**HallsofIvy** · November 15th 2012, 06:32 AM

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

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

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