${{\left| \overrightarrow{a}+\overrightarrow{b} \right|}^{2}}-{{\left| \overrightarrow{a}-\overrightarrow{b} \right|}^{2}}=\left( {{\left| \overrightarrow{a} \right|}^{2}}+{{\left| \overrightarrow{b} \right|}^{2}}+2\overrightarrow{a}.\overrightarrow{b} \right)-\left( {{\left| \overrightarrow{a} \right|}^{2}}+{{\left| \overrightarrow{b} \right|}^{2}}-2\overrightarrow{a}.\overrightarrow{b} \right)=$

$4\overrightarrow{a}.\overrightarrow{b}\xrightarrow{{{\left| \overrightarrow{a}-\overrightarrow{b} \right|}^{2}}\ge 0}{{\left| \overrightarrow{a}+\overrightarrow{b} \right|}^{2}}\ge 4\overrightarrow{a}.\overrightarrow{b}\xrightarrow{{{\left| \overrightarrow{a}+\overrightarrow{b} \right|}^{2}}=8}4\overrightarrow{a}.\overrightarrow{b}\le 64\Rightarrow \overrightarrow{a}.\overrightarrow{b}\le 16$ 

تذکر: حالت تساوی زمانی برقرار است که دو بردار $\overrightarrow{a}$ و $\overrightarrow{b}$ هم‌راستا، هم‌جهت و هم‌اندازه $\left( \left| \overrightarrow{a} \right|=\left| \overrightarrow{b} \right|=4 \right)$ باشند.