با توجه به شکل داریم:

$\overrightarrow{a}+(-\overrightarrow{b})+\overrightarrow{c}+(-\overrightarrow{d})=\overrightarrow{o}\Rightarrow \overrightarrow{a}+\overrightarrow{c}=\overrightarrow{b}+\overrightarrow{d}$

$\Rightarrow {{\left| \overrightarrow{a}+\overrightarrow{c} \right|}^{2}}={{\left| \overrightarrow{b}+\overrightarrow{d} \right|}^{2}}\Rightarrow {{\left| \overrightarrow{a} \right|}^{2}}+{{\left| \overrightarrow{c} \right|}^{2}}+2\overrightarrow{a}.\overrightarrow{c}={{\left| \overrightarrow{b} \right|}^{2}}+{{\left| \overrightarrow{d} \right|}^{2}}+2\overrightarrow{b}.\overrightarrow{d}$

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

$\Rightarrow 2(\overrightarrow{a}.\overrightarrow{c}-\overrightarrow{b}.\overrightarrow{d})={{2}^{2}}+{{4}^{2}}-({{1}^{2}}+{{3}^{2}})=10\Rightarrow \overrightarrow{a}.\overrightarrow{c}-\overrightarrow{b}.\overrightarrow{d}=5$