# Verify the parallelogram law for vectors u and v in

Verify the parallelogram law for vectors u and v in ${R}^{n}:\parallel u+v{\parallel }^{2}+\parallel u-v{\parallel }^{2}=2\parallel u{\parallel }^{2}+2\parallel v{\parallel }^{2}$

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alexandrebaud43

$\parallel u+v{\parallel }^{2}+\parallel u-v{\parallel }^{2}$
$=\left(u+v\right)\cdot \left(u+v\right)+\left(u-v\right)\cdot \left(u-v\right)$
$=\left(u\cdot u+v\cdot u+u\cdot v+v\cdot v\right)+\left(u\cdot u-v\cdot u-u\cdot v+v\cdot v\right)$
$=2u\cdot u+2v\cdot v$
$=2\parallel u{\parallel }^{2}+2\parallel v{\parallel }^{2}$
Hint:
$\parallel u{\parallel }^{2}=u\cdot u$

###### Not exactly what you’re looking for?
enlacamig

Let $u,v\in {R}^{n}$. Then we have
$\parallel u+v{\parallel }^{2}=\left(u+v\right)\cdot \left(u+v\right)$
$=\parallel u{\parallel }^{2}+2\left(u\cdot v\right)+\parallel v{\parallel }^{2}$
Also,
$\parallel u-v{\parallel }^{2}=\left(u-v\right)\cdot \left(u-v\right)$
$=\parallel u{\parallel }^{2}-2\left(u\cdot v\right)+\parallel v{\parallel }^{2}$
Therefore,
$\parallel u+v{\parallel }^{2}+\parallel u-v{\parallel }^{2}=\parallel u{\parallel }^{2}+2\left(u\cdot v\right)+\parallel v{\parallel }^{2}+\parallel u{\parallel }^{2}-2\left(u\cdot v\right)+\parallel v{\parallel }^{2}$
$2\parallel u{\parallel }^{2}+2\parallel v{\parallel }^{2}$