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

\(\displaystyle{\left({u}+{v}\right)}\times{\left({u}−{v}\right)}={u}\times{u}−{u}\times{v}+{v}\times{u}−{v}\times{v}={\left(∥{u}∥^{{{2}}}\right)}−{u}\times{v}+{u}\times{v}−{\left(∥{v}∥^{{{2}}}\right)}={\left(∥{u}∥^{{{2}}}\right)}−{\left(∥{v}∥^{{{2}}}\right)}={\left(∥{v}∥^{{{2}}}\right)}−{\left(∥{v}∥^{{{2}}}\right)}={0}\)

\(\displaystyle{\left({u}+{v}\right)}\times{\left({u}−{v}\right)}={u}\times{u}−{u}\times{v}+{v}\times{u}−{v}\times{v}={\left(∥{u}∥^{{{2}}}\right)}−{u}\times{v}+{u}\times{v}−{\left(∥{v}∥^{{{2}}}\right)}={\left(∥{u}∥^{{{2}}}\right)}−{\left(∥{v}∥^{{{2}}}\right)}={\left(∥{v}∥^{{{2}}}\right)}−{\left(∥{v}∥^{{{2}}}\right)}={0}\)