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

Question
Equations and inequalities
asked 2021-01-08
Prove that if \(\displaystyle{\left|{\left|{u}\right|}\right|}={\left|{\left|{v}\right|}\right|},{t}{h}{e}{n}{\left({u}+{v}\right)}·{\left({u}−{v}\right)}={0}.\)

Answers (1)

2021-01-09
Suppose u,v are two vectors in a real inner product space V such that \(\displaystyle∥{u}∥=∥{v}∥.\) Then, from the properties of the inner product (i.e. dot product), we see that
\(\displaystyle{\left({u}+{v}\right)}\times{\left({u}−{v}\right)}={u}\times{u}−{u}\times{v}+{v}\times{u}−{v}\times{v}={\left(∥{u}∥^{{{2}}}\right)}−{u}\times{v}+{u}\times{v}−{\left(∥{v}∥^{{{2}}}\right)}={\left(∥{u}∥^{{{2}}}\right)}−{\left(∥{v}∥^{{{2}}}\right)}={\left(∥{v}∥^{{{2}}}\right)}−{\left(∥{v}∥^{{{2}}}\right)}={0}\)
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