# Prove that if ||u||=||v||, then (u+v)·(u−v) = 0.

Prove that if $||u||=||v||,then\left(u+v\right)·\left(u-v\right)=0.$
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StrycharzT
Suppose u,v are two vectors in a real inner product space V such that $\parallel u\parallel =\parallel v\parallel .$ Then, from the properties of the inner product (i.e. dot product), we see that
$\left(u+v\right)×\left(u-v\right)=u×u-u×v+v×u-v×v=\left(\parallel u{\parallel }^{2}\right)-u×v+u×v-\left(\parallel v{\parallel }^{2}\right)=\left(\parallel u{\parallel }^{2}\right)-\left(\parallel v{\parallel }^{2}\right)=\left(\parallel v{\parallel }^{2}\right)-\left(\parallel v{\parallel }^{2}\right)=0$